Caribbean island exports what French drink (7)
I believe the answer is:
martini
'drink' is the definition.
(martini is a kind of drink)
'caribbean island exports what french' is the wordplay.
'caribbean island' becomes 'martinique' (island of the Lesser Antilles).
'exports' indicates named letters should be taken 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 "Italian drink" , "James Bond's drink of choice" , "Cocktail made with gin or vodka and vermouth" , "Type of Italian vermouth" , "Sienese painter" .)