Green’s Dictionary of Slang

take the mickey (out of) v.

also take a mike (out of), ...the mick (out of), …the Mickey Bliss, ...the micky, ...the mic-mac, ...the mike (out of)
[the assumed ety. is rhy. sl.: take the mickey bliss = take the piss (out of) under piss, the phr.; however, note that citations for the non-rhyming take a/the mike exist from 1920s, while those for take the mickey begin only in mid-1950s]

[1920s+] to tease; thus mickey-take, mickey-taking, mickey-taker (cf. extract the Michael v.).