Caribbean island exporting what French drink (7)
I believe the answer is:
martini
'drink' is the definition.
(martini is a kind of drink)
'caribbean island exporting what french' is the wordplay.
'caribbean island' becomes 'Martinique' (island of the Lesser Antilles).
'exporting' is a deletion indicator (to export can mean to send something away).
'what french' becomes 'que' ('what' in French).
'martinique' with 'que' taken away is 'MARTINI'.
(Other definitions for martini that I've seen before include "obtained from a bottle" , "James Bond's drink of choice" , "alcoholic drink" , "In glass, one's dry" , "Cocktail of gin and dry vermouth" .)