Stylish Hilary for one maintains business link? (8)
I believe the answer is:
intercom
'link?' is the definition.
(communication link)
'stylish hilary for one maintains business' is the wordplay.
'stylish' becomes 'in'.
'hilary for one' becomes 'term' (Hilary term is a term at the University of Oxford).
'maintains' is an insertion indicator.
'business' becomes 'co' (short for company).
'term' placed around 'co' is 'tercom'.
'in'+'tercom'='INTERCOM'
(Other definitions for intercom that I've seen before include "Phone system inside building" , "short range phone" , "System of communication often on door" , "Short-range telephone system" , "Communication device" .)