Cold month with early closing in holiday island (5)
I believe the answer is:
capri
'holiday island' is the definition.
'cold month with early closing in' is the wordplay.
'cold' becomes 'c' (eg on taps).
'month' becomes 'apr' (abbreviation for April).
'with early closing' means to remove the last letter (the word ends earlier than it normally would).
'in' with its final letter removed is 'i'.
'c'+'apr'+'i'='CAPRI'
(Other definitions for capri that I've seen before include "Old Ford model - Italian island" , "Island near Naples" , "Tourist destination" , "popular resort" , "Italian isle" .)