Belief unacceptable across island (6)
I believe the answer is:
notion
'belief' is the definition.
('notion' can be a synonym of 'belief')
'unacceptable across island' is the wordplay.
'unacceptable' becomes 'noton' (I've seen this before).
'across' means one lot of letters goes inside another.
'island' becomes 'i'.
'noton' placed around 'i' is 'NOTION'.
(Other definitions for notion that I've seen before include "Vague idea or opinion" , "Whimsical idea or small sewing article" , "clue" , "Idea, belief" , "Vague or whimsical idea" .)