1620s, "rice plant," from Malay (Austronesian) padi "rice in the straw." Main modern meaning "rice field, ground where rice is growing" (1948) is a shortening of paddy field.
"an Irishman," 1780, slang, from the pet form of the common Irish proper name Patrick (Irish Padraig). It was in use in African-American vernacular by 1946 for any "white person." Paddy-wagon is attested by 1930, perhaps so called because many police officers were Irish. Paddywhack (1811) originally meant "an Irishman;" with the second element apparently added vaguely for emphasis.