 Following the war, Baker made a single appearance in first-class cricket when he was selected to play for the Royal Navy against Cambridge University at Fenner's in 1920.Batting twice in the match, he was dismissed for 6 runs by Charles Marriott in the Royal Navy first-innings, while in their second-innings he was dismissed for 28 runs by Gilbert Ashton.Baker's parents were married in 1881 in New Orleans; Arthur Conyers Baker was given as the seventh son of Colonel George Baker, 16th Lancers, and marrying Mary Abercromby Clayton, daughter of Dr John Clayton of Banff.