Proprietor gets fat around island (5)
I believe the answer is:
laird
'proprietor' is the definition.
(thesaurus)
'fat around island' is the wordplay.
'fat' becomes 'lard' (I've seen this before).
'around' means one lot of letters goes inside another.
'island' becomes 'i'.
'lard' placed around 'i' is 'LAIRD'.
'gets' is the link.
(Other definitions for laird that I've seen before include "Scottish estate-owner" , "Scottish owner of large estate" , "Scottish squire" , "Scot. landowner" , "Big landowner" .)