Animals, and people, can and have continued to be alive and… somewhat, or even mostly functional… with portions of their brains obliterated or removed.

Go look up Phinneas Gage, or the history of lobotomies and brain surgeries.

Now, this person could be describing bloody chunks of the skull and skin amd fur, but it is not strictly impossible that the deer could have actually managed to loose chunks of its actual brain, and still remain capable enough to keep functioning…

… for a while, at least. If it did really cave in its own skull, it would almost certainly die from cranial hemmoraghing soon afterward, if not that, infections.