Drink making dame drunk (4)
I believe the answer is:
mead
'drink' is the definition.
(mead is a kind of drink)
'dame drunk' is the wordplay.
'drunk' indicates an anagram.
'dame' with letters rearranged gives 'MEAD'.
'making' is the link.
(Other definitions for mead that I've seen before include "Alcoholic drink made with honey" , "Cornish drink" , "something to drink" , "The poetic field" , "Cooked joint" .)