The idiom is on the lam, not on the lamb. The exact origins of this sense of lam are unknown, but it’s believed to be a late 19th-century U.S. slang term. It was originally a verb meaning to escape, and it’s still occasionally used in that sense, but today it mostly functions as a noun. To be on the lam is to escape, to flee justice, or to be in hiding from law enforcement.
An accused East Coast mobster, on the lam for more than 10 years, was found by the FBI in an unlikely occupation and location. [New York Daily News]
Mr. Israel, who went on the lam briefly, is currently serving a 22-year prison sentence. [Wall Street Journal]
Brosnan then attempts to pin the killing on the shallow, silly, former Dead Head Vanderveer who is forced to go on the lam to clear himself. [CultureMap]