Former lover lives on French island following banishment (5)
I believe the answer is:
exile
'banishment' is the definition.
(exile is a kind of banishment)
'former lover lives on french island' is the wordplay.
'former lover' becomes 'ex' (ex-lover).
'lives on french island' becomes 'ile' (I can't justify this - if you can you should give a lot more credence to this answer).
'ex'+'ile'='EXILE'
'following' is the link.
Can you help me to learn more?
(Other definitions for exile that I've seen before include "Person banished from the home country" , "Throw out" , "Ban to abroad" , "Banish from one's country" , "Expulsion from one's country" .)