I'm fed up with the wet country (7)
I believe the answer is:
bahrain
'country' is the definition.
(Bahrain is an example)
'i'm fed up with the wet' is the wordplay.
'i'm fed up' becomes 'bah' (old-fashioned expression of disapproval).
'with' means one lot of letters go next to another.
'the wet' becomes 'rain' (the wet can mean wet weather).
'bah'+'rain'='BAHRAIN'
(Other definitions for bahrain that I've seen before include "Arab state" , "Arab sheikhdom" , "Gulf state" , "kingdom in Middle East" , "Gulf region?" .)