Yes, the rest of the body could still function.

That said, it probably wouldn't function for that long. The brain has "autonomic" functions that help maintain proper blood pressure, etc. After it's gone, the body will eventually become hemodynamically unstable and cardiac failure is pretty much inevitable. I suppose some future medical advances might one day prevent that. But really, why bother?

In practice, brain death is legal death. Hospitals will not provide life support to a dead person, unless they are an organ donor. So once brain death is diagnosed, one way or another that person will soon be in the morgue.