Avoid our island's river (8)
I believe the answer is:
missouri
'river' is the definition.
(Missouri is an example)
'avoid our island's' is the wordplay.
'avoid' becomes 'miss' (miss can mean to avoid).
'island' becomes 'i' (geographical abbreviation).
'miss'+'our'+'i'='MISSOURI'
(Other definitions for missouri that I've seen before include "24th American state to join the Union, 1821" , "Longest river in the US, joining the Mississippi at St Louis" , "Flower" , "State of central US" , "State south of Iowa" .)