Carpenter collected a couple of examples in 1929, but each source claims to have learned the song some 75 years earlier: i.e. in the early 1850s (which may possibly point to a minstrel origin).

First line - "I lost my jacket in the alley" from Edward Robinson, of Sunderland. Born 1834, to sea 1846. Carpenter's notes say Captain Page heard the chantey about 1853. (He collected from both Capt Mark Page and Capt Edward Robinson in the same retired sailor's home in Sunderland, which may explain the ambiguity of the note)

First Line - "O, I lost my coat in Story's Alley" from James Forman, of Leith, born 1844, first ship 1856. "Learned as a boy before going to sea" and a further note "Story's Alley in Leith".

I'm not at home at present, but at least one version (Forman?) is on Kennedy's selction of Carpenter recordings, and the chorus sounds very much like "I'm Billy in the army"