Bear is swimming around island in a chilly part of the world (7)
I believe the answer is:
siberia
'a chilly part of the world' is the definition.
(extremely cold Russian region)
'bear is swimming around island' is the wordplay.
'swimming' indicates anagramming the letters.
'around' means one lot of letters goes inside another.
'island' becomes 'i'.
'bear'+'is'='bearis'
'bearis' anagrammed gives 'sibera'.
'sibera' going around 'i' is 'SIBERIA'.
'in' acts as a link.
(Other definitions for siberia that I've seen before include "Region of Russia, from the Urals to the Pacific, famously cold in winter" , "Large E. Russian region" , "Russia's far east" , "Arctic region" , "Area of eastern Russia" .)