Caribbean island bans that Parisian cocktail (7)
I believe the answer is:
martini
'cocktail' is the definition.
(martini is a kind of cocktail)
'caribbean island bans that parisian' is the wordplay.
'caribbean island' becomes 'martinique' (island of the Lesser Antilles).
'bans' indicates named letters should be taken away.
'that parisian' becomes 'que' ('that' in the French language).
'martinique' with 'que' taken out is 'MARTINI'.
(Other definitions for martini that I've seen before include "Italian drink" , "obtained from a bottle" , "Sienese painter" , "Cocktail of gin and dry vermouth" , "Cocktail made with gin or vodka and vermouth" .)