Gave up on dress designer (8)
I believe the answer is:
resigned
'gave up' is the definition.
('resign' can be a synonym of 'give up')
'dress designer' is the wordplay.
'dress' indicates anagramming the letters.
'designer' is an anagram of 'RESIGNED'.
'on' acts as a link.
(Other definitions for resigned that I've seen before include "Defeatist" , "Long-suffering" , "Abdicated" , "without hope of change" , "Voluntarily stood down" .)