Crush their throat. Just jab the finger into the front of the throat as hard as you can.

It's very easy to produce serious injury and death there although you can expect your opponent to try and protect the area.

You may get better results by folding the finger and striking with the 2nd joint rather than using the finger fully extended. You will get a slightly larger striking surface but a much stronger one. I suggest practicing on some disposable peasants before using it on a real target.

Hard training of single digits

Take inspiration from this fella from Malaysia. He trained (and often broke) his right index finger to the point where he is able to pierce coconuts. Such a blow to a human torso could result in lethal injuries. I imagine, that upon passing of this law, many nobles start training one of their fingers to this extent. No magic or superpowers required.

To add on Ash's answer, Vagus nerve stimulation can lead to collapse and even to death, as consequence of induced bradycardia.

Parasympathetic innervation of the heart is partially controlled by the vagus nerve and is shared by the thoracic ganglia. Vagal and spinal ganglionic nerves mediate the lowering of the heart rate. The right vagus branch innervates the sinoatrial node. In healthy people, parasympathetic tone from these sources are well-matched to sympathetic tone. Hyperstimulation of parasympathetic influence promotes bradyarrhythmias. When hyperstimulated, the left vagal branch predisposes the heart to conduction block at the atrioventricular node.

Sometimes even a too tight necktie can achieve that result.

Poison

I understand that there is a rather fatal exotic poison that can be applied to the fingertip.

The finger then may be either jabbed in the opponent's eye or his mouth, or if you're feeling like making a particular point, up his nose.

Carefull to not get cut during the fight though, the poison may find it's way into your bloodstream.

Maybe the mouth's not such a great idea as a target, there being teeth after all.
The most suspicious and canny of opponents would wear a nasal insert with a downward facing pin - beware. No doubt the referees are wise to this.

The most dishonourable and disgraceful way to die is the old anal-finger-poisoning (familiar from Brazillian Ju-Jitsu), not for civilised opponents.

Sir Barrard probably could kill him with one blow, in theory at least. In particular the hyoid bone is surprisingly delicate and can be crushed with a sharp blow, a single finger could be sufficient to do the damage. It might also be possible to do enough damage to the carotid or better yet, because it's closer to the surface, the jugular to cause death after rather than during the fight due to internal hemorrhage. The brachial artery is also close enough to the surface that it may be vulnerable to such damage, not immediately but eventually deadly. Fragmentation of the nasal bones could also be lethal, but would almost certainly require multiple strikes as they are much stronger.

In my imagining, these are the sorts of disputatious nobles who wear ruffled shirts, powdered hair with little pig tails out the back, short breetches, shoes with buckles, codpieces, all that sort of finery. Think of a men writing with feathers mgm movie from the 30s, such as the scarlet pimpernel - only much, much worse.

In preparation for the finger fight, legs and non-participating arm are tightly bound, only the belligerent finger and arm left unrestrained. They are also blind-folded. A string quartet provides background music, claret is served to the spectators, who view from a raised balcony above the arena.

When the magistrate sounds action, the disputants are free to bound around the place, flinging their fingers furiously, un a d'autre (that's french).

Each combatant has a receptor implanted somewhere upon their person which when actuated with a firm jab, will release a dose of immediately fatal poison.

And so, the spectacle becomes a series of furious jabbing exchanges where, as with the gladiators of old, points are awarded for style. The woman play the closest attention.

If, after a suitable interval of say 20 minutes or so, no one is yet dead, the survivors are both lauded as the best of all posible men, all trivial injury forgiven, any serious injury recompensed, and the wine flows freely.

If one is killed, the wine still flows freely, with perhaps a tear for the vanquished, or perhaps not, depending.

He could possible fatally wound another person but i would suggest that instead of his index finger he use his thumb. By curling his fist and placing his thumb along the top of his index finger like this picture below

In Karate I believe this is called a Boshiken or thumb strike. I believe it can be used to dig into various pressure points around the head and the throat and can even be used to gouge out eyes it also has the effect of allowing you the thumb to hit even a hardened part of the body with out touch much damage and does not require some of the more long term ruinous contact conditioning that say training something like the Nukite or spear hand can cause.
in this instance the contact point will still be a single digit just not a finger.

Other answers already suggested attacking the throat area, which indeed has a number of fatal vulnerabilities. However, even inexperienced duelist would know to keep his throat protected, and attacker would more likely just break his finger trying to push through defensive blocks.

Martial arts like Kyusho Jitsu can provide an idea on how finder-fighting duels can unfold. An attacker can strike a number of vulnerable points in the body, which would cause the defender to drop his defense, at least momentarily.

More experienced defenders would try to break attacker's finger with powerful elbow or forehead blocks.

There are many places on the body where a single finger can produce fatal injuries and it is impossible to cover them all simultaneously.

Everyone has already mentioned the throat, there are a number of sensitive spots there that are easily damaged.

The solar plexus is a nerve cluster that can cause fatal paralysis of the diaphragm if struck with sufficient force.

Near the solar plexus is the xiphoid process; a little tab of bone which, if broken off, often punctures the liver and results in fatal internal bleeding.

The temple is another potential soft spot, being the thinnest point in the skull. Most techniques strike at it with an elbow, but a sufficiently sturdy finger could easily do the job as well.

And finally, if he wants to go for maximal vomiting amongst the audience, just put an infectious agent under the fingernail and jab straight through the eye. There's a decent chance he can damage the blood vessels behind the eye and cause them to bleed into the brain causing eventual death, and if not the infection is likely to be fatal anyway unless their doctors are really good and have antibiotics. Even if he doesn't kill his opponent he will have permanently disfigured him.

Additionally, a sufficiently skilled martial artist could make the killing blow look like an accident. Stumble, trip, everyone lands really hard and whoops, somehow there's a broken neck. What are the odds?

Sir Barrard uses his pinky finger.

Considered as the weakest finger of all, using the pinky will surprise or even destabilize his opponent: Sir Barrard surely is strong to have confidence in this finger! But the trick is there: the pinky finger is in the direct continuity of the side palm of the hand, and if it is not a good thrusting weapon, it can reveal a great slashing weapon.

Now, with all his might, Sir Barrard must wait for an oppurtunity to deliver a slashing uppercut with his little finger upward the nose of his opponent, the idea being breaking his nose and sending pieces of the cartilage directly into the skull.

Source: Quora

Note that the blow must me delivered with a tremendous force, and this will likely result in Sir Barrard's finger and metacarpals being broken in the process.

In real life, delivering such a blow and kill even with a clenched fist is hard to manage. But this is fiction, and you can use artistic license as long as it fits the plot in a spectacular way...

Death yoga.

source

The chakras are yogic energy points within the body. The points govern and energize energy flows. One hears commonly about manipulating or otherwise focusing the various chakras, usually to achieve desired beneficial outcomes for the practitioner. Examples:

http://www.soulandspiritmagazine.com/13951-2/

Crown (Seventh) Chakra

Located on the top of the head, the crown chakra symbolises
spirituality and your connection to a higher power...

Tapping point: top of the head

Third Eye or Brow (Sixth) Chakra

This chakra is located directly between the eyebrows. It represents
intuition, imagination, reflection, and the ability to see things for
what or how they are. ...

Tapping point: between the eyebrows (be extra gentle with this point)

Sacral (Second) Chakra

The sacral chakra, also referred to as the womb chakra, is located in
the pelvis behind the navel. It relates to your creativity and
feelings and is also linked to childlike joy...

Tapping point: just below the belly button

Root (First) Chakra

The root chakra is located at the base of the spine. It represents
your feelings of safety and survival. It deals with issues of
abandonment, unworthiness, and insecurity. Its focus is safety,
security, and survival.

Tapping point: lower sacrum or top of your thighs.

There is more to these energies than this simple synopsis. The energy flows within the body can be and possibly must be connected to the larger world.

Energy is power: neither good not bad. Energy that can be used to heal can be used to harm. Your fingerfights are offensive yoga - the goal being to destructively manipulate your opponents energies in the context of the larger energy of the world.

Note:yogic manipulation of your opponents energy can kill, but it might not kill right away. The vibrations set in motion may take time to resolve into a pattern that causes death. Other things may be happening in your opponent over this time which he might be able to resolve on his own or with help - or which he might use to his own ends.

I did not make this up.
https://en.wikipedia.org/wiki/Touch_of_Death

If you are willing to bend the rules a bit, you could take inspiration from Naruto vs Neji fight.

In the Naruto universe, every person has many Chakra/Pressure points throughout their body. The chakra points release chakra/energy that is essential for the body to function. If you close off a chakra point, then the corresponding body part stops functioning (or at least becomes much weaker). In this fight, Neji slowly closes off all of Naruto's chakra points, and as a result, Naruto's body becomes so weak that he can barely even stand.

In your world:

• Maybe the knowledge about pressure points is not common and Sir Barrard incapacitates his baffled opponent one pressure point at a time.




• Or it is common knowledge in dueling circles, and it comes down to the skill of the fighters -- who can close off more/important pressure points first and incapacitate/weaken the other.

Once the opponent is sufficiently incapacitated, and their defenses are weakened, a single powerful jab to the throat can be enough to finish them off.

Plus, the second method makes the fights a lot more interesting to watch than fighters just trying to poke each other in the eyes/throat/groin.

Depending on how you define "unsportsmanlike" behaviour, I would say that the answer is probably a rabbit punch. As long as you are allowed to strike with the side of your finger and not just the tip, you can deliver a significant amount of force with a chop, either on the outer edge of the little finger or the edge of the thumb.

A rabbit punch is a sharp strike to the back of the neck/base of the skull. It is called a rabbit punch not because it's small or ineffectual, but because it's a quick and clean way to kill a rabbit. It can separate the brain from the spinal cord, which is certainly fatal.