Island's cold - month curtailed (5)
I believe the answer is:
capri
'island's' is the definition.
(Italian island)
'cold month curtailed' is the wordplay.
'cold' becomes 'c' (eg on taps).
'month' becomes 'april' (April is an example).
'curtailed' means to remove the last letter.
'april' with its last letter removed is 'apri'.
'c'+'apri'='CAPRI'
(Other definitions for capri that I've seen before include "Old Ford model - Italian island" , "Neapolitan isle" , "Mediterranean island" , "Romantic and expensive island in Bay of Naples" , "Italian isle" .)