Green’s Dictionary of Slang

(US drugs) marijuana; a marijuana cigarette, esp. one of poor quality.

1953		W. Brown Monkey On My Back (1954) 93: Bammies (the poorest grade of marijuana cigarettes) could be bought for as little as three for a quarter.
1956		E. Hunter Second Ending 302: When you’re hooked, you’re dead. It takes a lot of guts [...] more guts than you were thinking of when you grabbed that cute little bammie. When you are dead, there is nothing but heroin.
1959		W. Brown Teen-Age Mafia 52: Crummy music. Bammies that are mostly hay. Beer at a buck a throw.
2001		ONDCP Street Terms 2: Bammy — Marijuana.