Live inside new part of mosque (7)
I believe the answer is:
minaret
'part of mosque' is the definition.
('minaret' is part of 'mosque')
'live inside new' is the wordplay.
'live' becomes 'are' ('they live' means 'they are').
'inside' is an insertion indicator.
'new' becomes 'mint ' (I've seen this before).
'are' inserted within 'mint' is 'MINARET'.
(Other definitions for minaret that I've seen before include "tower in Istanbul, say" , "Slender mosque tower" , "Slender tower with balconies" , "Tower from which muezzin calls Muslims to prayer" , "Tower attached to mosque" .)