Landowner gets fat around island (5)
I believe the answer is:
laird
'landowner' is the definition.
(Scottish term for a landowner)
'fat around island' is the wordplay.
'fat' becomes 'lard' (I've seen this before).
'around' is an insertion indicator.
'island' becomes 'i'.
'lard' placed around 'i' is 'LAIRD'.
'gets' is the link.
(Other definitions for laird that I've seen before include "Big landowner" , "Scottish squire" , "Scots estate owner" , "Scottish owner of large estate" , "Scottish estate-owner" .)