Give up completely? Not completely (4)
I believe the answer is:
quit
'give up' is the definition.
('quit' can be a synonym of 'give up')
'completely? not completely' is the wordplay.
'completely?' becomes 'quite' (I've seen this before).
'not completely' means to remove the last letter.
'quite' with its last letter taken away is 'QUIT'.
(Other definitions for quit that I've seen before include "Pay off (a debt)" , "Vacate" , "Leave or abandon" , "Leave office" , "Leave; give up" .)