Girl, in fact, needing no introduction (4)
I believe the answer is:
ruth
'girl' is the definition.
'fact needing no introduction' is the wordplay.
'fact' becomes 'truth' (truth is a kind of fact).
'needing no introduction' means to remove the first letter.
'truth' with its first letter taken away is 'RUTH'.
'in' acts as a link.
(Other definitions for ruth that I've seen before include "once felt for a Moabite woman" , "Lady" , "- - Rendell, crime writer" , "-- Rendell, novelist" , "Biblical character" .)