It would be exceedingly difficult to extract the brain and keep it alive. People think that the brain is a separated thing. It isn’t. It’s an integrated part of the nervous system, of a person.

To separate the brain, you need to cut the brain stem somewhere. Because the brain stem regulates vital functions like heart rate and breathing, so you would want to leave that part with the body. It would be very difficult to separate the brain from the brain stem without destroying some vital part.

You would not be able to attach the brain to the nervous system again. But for the question, to keep the brain alive. Yes, kind of, with a good blood and liquor (cerebrospinal fluid) supply, at least for a while. It’s unknown if the brain would be conscious, in excruciating pain, or unconscious. But If you separate the brain stem from the brain, the brain will likely become unconscious, because the brain stem also regulates the level of consciousness, and initiates it.

Below, a dissected brain, including a big part of the nervous system.