Green’s Dictionary of Slang

1. a mad or eccentric person.

1912		Shaw Pygmalion II: He’s off his chump, he is. I dont want no balmies teaching me.
1919		Star (Wshington, DC) 4 May 4/5: ‘Buck up, Balmy,’ I said.
1943		J. Phelan Letters from the Big House 38: He don’t think he ain’t never coming in stir, oh no. No one don’t. Especial balmies.
1954		J. Phelan Tramp at Anchor 160: A balmy is a man who is prepared and even anxious to short-circuit the many little evasions and pretences of civilized life.

2. a prison wing for disturbed inmates.

1943

J. Phelan Letters from the Big House 37: The balmy’s in ‘M’ Hall in them days.