Live like a headless chauvinist (5)
I believe the answer is:
exist
'live' is the definition.
(I know that live can be written as exist)
'headless chauvinist' is the wordplay.
'headless' means to remove the first letter.
'chauvinist' becomes 'sexist' (I've seen this in another clue).
'sexist' with its first letter removed is 'EXIST'.
'like a' is the link.
(Other definitions for exist that I've seen before include "Subsist" , "Get a life!" , "Have objective reality" , "quick!" , "Survive" .)