Island held by fat estate owner (5)
I believe the answer is:
laird
'estate owner' is the definition.
(I know that laird is a type of large estate owner)
'island held by fat' is the wordplay.
'island' becomes 'i' (geographical abbreviation).
'held by' means one lot of letters goes inside another.
'fat' becomes 'lard' (lard is a kind of fat).
'i' placed within 'lard' is 'LAIRD'.
(Other definitions for laird that I've seen before include "Scottish squire" , "Estate landowner" , "Man owning land" , "Proprietor" , "Scottish owner of large estate" .)