Green’s Dictionary of Slang

(Aus./US) to treat to a drink; thus ball-em-off, one who buys a round.

1847		Bell’s Life in Sydney 20 Mar. 2/6: The roudy landlord was [...] extracting the loose pewter from the kicks of the heterogenous assembly of ball-em-offs.
1892		Brooklyn Dly Eagle (NY) 4 Dec. 3/7: It was nothing new for ‘de gang’ to be dead broke, but [...] buoyed up with the hope that one of their number would strike luck an generously ‘ball off’ the crowd.