If your idea of self-defense is last option, close-up and personal (as opposed to defense-of-the-free-world gunfighting) a .22LR or, better, .22WMR will suffice. Few will survive a shot under the chin, in the eye, in the throat or up the nose.

Is it optimal? No. Winter clothes will all but totally defeat it. But, at mugging distances... it will suffice. As others have pointed out... the .22 can make an ugly wound.