Can a brain be transplanted?
"Is it possible" has a carefully conditional "yes".  Do we have the expertise or technology to do it now? No.  One of the biggest difficulties is that nerve tissues (the connections between brain and rest of body) will take time to heal... time during which the brain cannot sufficiently give commands to the rest of the body to keep autonomic functions running.

Head transplants, however, have been "successfully" completed using monkeys.  The monkey lived for sometime after the transplant, but eventually died.  China has also been known to do a similar procedure with dogs.

Caution, this may be too explicit for some; I'm not sure ethics boards would allow it nowadays:
Monkey Head Transplant