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 and 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.
Ah yes, I heard about that one when I was a kid, forgot to mention it!
Damn headless chicken stayed alive for weeks... months I think?
It would walk around attempting to peck at things with a head it didn't have... they used a little dropper to squirt some mushed up food ... directly into its throat...
...and it actually died because one day it choked during this kind of feeding session, concievably it could have lived longer had that not happened.
Or at least thats the 'death account' I remeber my grandpa telling me, apparently there are multiple stories as to what exactly cause Mike to pass on.
Yeah, that is why I left a little wiggle room in my statements. But coordinated movements when the cranial cavity is so damaged that chunks of brain are spilling out? A deer's brain isn't even very large. There's hardly chunks to be had there. I dunno there's some chance that the deer was already dead and this is just some sort of postmortem spasms. But I am just doubtful over the whole thing as written.
Fair enough. If I witnessed a deer bashing its own brains out, I would definitely be so fucking freaked out my memory would be doing wild shit about the encounter, too. Filling in details that weren't there, not noticing that only part of the brain fell out, just completely unreliable.