Written and first recorded (as “Pussy”) by Harry Roy & His Bat Club Boys (1931).
Also recorded by R. Crumb & His Cheap Suit Serenaders (1978).
From the wiki: “Harry Roy was born Harry Lipman in Stamford Hill, London, England, and began the study of the clarinet and alto saxophone at the age of 16. He and his brother Sidney formed a band which they called The Darnswells, with Harry on saxophone and clarinet and Sidney on piano. During the 1920s they performed in several prestige venues such as the Alhambra and the London Coliseum, under names such as The Original Lyrical Five and The Original Crichton Lyricals. They spent three years at the Café de Paris, and toured South Africa, Australia and Germany.
“By the early 1930s, Harry Roy was fronting the band under his own name, and broadcasting from the Café Anglais and the Mayfair Hotel. In 1931 he wrote and sang ‘My Girl’s Pussy’, which has since been the subject of many cover versions and remakes, including a novelty 78-rpm 10-inch single released in 1980 by R. Crumb & His Cheap Suit Serenaders (after its 1978 appearance on R. Crumb and his Cheap Suit Serenaders No. 3 long-play album).”