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I need another excuse of "I was very busy" as people became tired of hearing it. So, I thought of expressing the idea of having a very restricted/limited time for all the tasks that I have been assigned to do and therefore I couldn't completely finish this specific one. The thing is I can't get my hands on a suitable phrase.

I guess it would be something like:

I wasn't able to finish it as I got pretty restricted in time.

Is it correct? Can I achieve that meaning better?

It is not necessary to be formal, but I don't want it to be too informal.

There are some other good suggestions here. I might also use:

I got caught up with other work.

I got caught up in other work.

Either of these means that you were very busy or “tangled up” with other work- like you couldn’t escape the trap of the work, so to speak.

EDIT: Sorry, I should have also addressed your own suggestion -

I got pretty restricted in time.

I would not say that, even if the meaning might be understood. The phrasing is awkward - it would sound better to say

My time was restricted.

if you were to use the word restricted. Restricted is something that is, not something that is gotten, generally speaking.

Also, restricted is often used to indicate some specific enforced limit, like "Access was restricted to authorized users", or "My time was restricted to three hours". I think in this case the word doesn't completely fit if nobody was actually forcing you to only spend a certain amount of time on this particular task. I daresay it might come off as rude to tell someone the time you had for this task was restricted; they might think you set a limit on the time you were going to spend on it because you didn't think it was that important.

Get around to
phrasal verb of get

deal with (a task) in due course.

to do something that you have intended to do for a long time

I didn't get around to putting all the photos in frames.

I couldn't get around to finishing it on time.

I intended to tidy the flat at the weekend, but I didn't get round to it.

It's been at the back of my mind to call José for several days now, but I haven't got round to it yet.

He never did get around to putting up the shelves.

After weeks of putting it off, she finally got around to painting the bedroom.

Did you get round to doing the shopping?

to be tied up

to be very busy and unable to speak to anyone, go anywhere, etc:

Fig. busy.

How long will you be tied up? I will be tied up in a meeting for an
hour.

I was tied up and couldn’t get to the phone.

He's tied up with his new book. He's working hard, you know.

To have too much on plate

to be too busy.

I'm sorry, I just have too much on my plate right now. If you have
too much on your plate, can I help?

You could've said:

I wasn't able to finish it, because I had too much on my plate.

or

I wasn't able to finish, because I was a little tied up.

or

I couldn't get around to it.

Suggestions:

I wasn't able to do it because I ran out of time.

There wasn't enough time to do everything I needed to do.

I didn't have sufficient time to do everything.