It’s commonly thought that take the mickey is rhyming slang, Mickey/Mike Bliss = Piss, but that may not be the case. The phrase take the mike is even older. From Green’s Dictionary of Slang:
1922 [UK] ‘J.H. Ross’ Mint (1955) 121: But mate, you let the flight down, when he takes the mike out of you every time.
1937 [UK] J. Curtis There Ain’t No Justice 11: Everyone round here takes the mike outa him just because he wears glasses and can’t talk without stammering, but my Ernie’s worth ten of your tu’penny ha’penny boxers.
1939 [UK] V. Davis Gentlemen of the Broad Arrows 117: Do you know what happens to saucy lags who try to take a mike out of me?
1941 [UK] G. Kersh They Die with Their Boots Clean 190: Are you trying to take the mike out of me? Or are you just potty?
The OED has:
a1935 T. E. Lawrence Mint (1955) ii. vi. 117 But, mate, you let the flight down, when he takes the mike out of you every time.
1935 ‘G. Ingram’ Cockney Cavalcade i. 14 He wouldn’t let Pancake ‘take the mike’ out of him.
1940 Notes & Queries 1 June 382/1 ‘Taking the mike out of’ anyone means pulling his leg, having a game with him.
Of course, take the piss could be older and unrecorded. It’s not a phrase that would make it into print in that earlier era.