Who's the earliest rock band to use a sitar on a song recording or album?
Dates are release dates.
Actual first appears to be the Beatles: "Norwegian Wood": December 1965.
but it followed a time of many musicians being interested in Indian music, influenced by Ravi Shankar who toured Europe and the USA in the late 1950s and early 1960s, sharing a record label with the Byrds. David Crosby (with The Byrds at that time), brought the music to the attention of George Harrison.
Influenced :- using sitar-like sound on guitar or similar, see the wiki article for more detail
Davey Graham: "She moved through the fair" 1963 Yardbirds: "Heart full of soul" : June 1965 : influenced, demos used sitar Kinks : "See my friends" : July 1965, imitated the sound with guitar