Be a headless chauvinist (5)
I believe the answer is:
exist
'be' is the definition.
(I know that be can be written as exist)
'a headless chauvinist' is the wordplay.
'a headless' means to remove the first letter.
'chauvinist' becomes 'sexist' (I've seen this in another clue).
'sexist' with its initial letter taken off is 'EXIST'.
(Other definitions for exist that I've seen before include "Subsist" , "quick!" , "Scrape by" , "Have objective reality" , "Have being or continue living" .)