Doctor has to remain in Island city (8)
I believe the answer is:
shanghai
'city' is the definition.
(Shanghai is an example)
'doctor has to remain in island' is the wordplay.
'doctor' indicates an anagram.
'to remain' becomes 'hang' (as in to hang around).
'in' indicates putting letters inside.
'island' becomes 'i'.
'has' is an anagram of 'sha'.
'sha' enclosing 'hang' is 'shangha'.
'shangha'+'i'='SHANGHAI'
(Other definitions for shanghai that I've seen before include "Kidnap in Chinese city-port?" , "How to impress a sailor" , "part of China?" , "Trick (someone) into doing something" , "many Chinese folk?" .)