In the Pacific, a private island (5)
I believe the answer is:
capri
'island' is the definition.
(Italian island)
'in the pacific a private' is the wordplay.
'in' says the answer is hidden in the clue.
'CAPRI' is hidden within 'pacific a private'.
(Other definitions for capri that I've seen before include "Location for holiday" , "Island home of G. 24" , "Island in the Gulf of Naples" , "popular resort" , "Island off Italy" .)