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Within my fantasy book I have a species of animal which lives in a dry climate - the desert. This animal is a furry mammal with four legs; for simplicity, let's say it looks like a bear. The species is thought to have gone extinct hundreds of years ago but my protagonist finds it as a baby on his journey.

What I want is an explanation or reason of how this animal could have survived. Is there an explanation - even a plausible magical one - by which this animal could have survived within the embryonic sac? I don't want it to be inside an egg and then hatch. But are there any other option/explanations without having it be hatched out of an egg?

The animal in question is a carnivore.

As a species living in a desert, it is possible that they have developed two key adaptations that assist in their survival in such a harsh environment.

The first, is Aestivation, a process by which an organism goes into a state of dormancy in order to survive extremes of heat or desiccation.

https://en.wikipedia.org/wiki/Aestivation

The second is Embryonic Diapause, a process by which an organism can but a pregnancy on hold until conditions are more conducive to rearing offspring.

https://en.wikipedia.org/wiki/Embryonic_diapause

I see the events as going as follows: The mother has been in Aestivation deep underground for a century. She was pregnant when she entered dormancy, and has kept her embryo in a state of diapause throughout. Eventually conditions improved and she emerged to the surface, only to find she was the last of her kind. She gave birth to her cub, but shortly afterward was killed by a predator, or even by people. Her cub survives.

Explanation 1

It never went extinct. People just thought it was extinct. Your character has stumbled upon a forgotten oasis where the remaining few of these creatures live.

Explanation 2

There are hidden underground caves where the creatures live. A sandstorm cleared one of the entrances to the cave system. The baby creature found its way out and is lost. Maybe the entrance has closed up again.

Explanation 3

This is the desert. Your character is suffering from dehydration and heatstroke. S/he is hallucinating and there is no bear there at all.

Unless I am misunderstanding the "not hatched from an egg" part, you are saying that it needs to be an actual live birth, except that its mother (like all others of its species) has been dead a long time.

There are some species that can exist in dormant states – microscopic creatures that can be frozen or dehydrated and later recover, hermetically sealed seeds and spores that can wait indefinitely before germinating, etc. – and I don't think there's any fundamental reason why that couldn't happen to a large pregnant animal.

So you could have the mother dehydrated, or in some kind of finely-tuned hibernation estivation, for centuries; and then she is revived by a monsoon or earthquake or whatever is required, and then she successfully gives birth but doesn't survive.

Two options come to mind, depending on how strict you want to be with interpreting "it was just born".

Option 1: It was aestivating. This is basically the desert-animal equivalent of cold-resistant frogs getting frozen in ice for the winter. Rather than freezing and thawing out, the creature just dries out, producing tons of special sugars to effectively glassify its cells and cease all life processes until a rainstorm or a flash-flood or something rehydrates it centuries later. As far as I know, there are no vertebrates, let alone mammals, that can aestivate that long or handle the concomitant extremes of dessication, but there is no fundamental biochemical reason why they couldn't. Plenty of reptiles and amphibians aestivate, and many invertebrates can handle extreme dessication.

Option 2: Its species has alternating generations of different morphs, or just a multi-form life cycle, like a cnidarian (or, heck, like pupating insects). It's juvenile / sessile form was aestivating; maybe that's even the purpose of the sessile form--to be a simpler organism that can more easily survive extreme conditions, and grow a motile form when conditions are favorable. And it just happened to take a few centuries before, as above, a flash flood or rainstorm or something re-activated it, and the motile form was literally just born from it.

Long ago, an adventurer was on a quest to kill some adults of the species. One of his companions found the freshly born cub nearby after the adults had been killed and put it into a magical time warp to save it from the guy who was on the quest. The magician intended to come back and free it, but he died not long after. The spell has weakened over the centuries and the cub came back into the normal world.