First public appearance of woman, extremely unfit (5)
I believe the answer is:
debut
'first public appearance' is the definition.
(I've seen this in another clue)
'woman extremely unfit' is the wordplay.
'woman' becomes 'deb' (deb is a kind of woman).
'extremely' suggests removing the centre (only the extremes of the word are used).
'unfit' with its centre taken out is 'ut'.
'deb'+'ut'='DEBUT'
'of' acts as a link.
(Other definitions for debut that I've seen before include "Initial appearance" , "Introduction" , "First appearence as of society girl" , "Tubed (anag.)" , "The coming out of a society girl" .)