American chases lost sheep in ancient city (7)
I believe the answer is:
ephesus
'ancient city' is the definition.
(ancient Greek city)
'american chases lost sheep' is the wordplay.
'american' becomes 'US' (abbreviation for United States).
'chases' is a charade indicator (letters next to each other).
'lost' indicates an anagram.
'sheep' is an anagram of 'ephes'.
'us' after 'ephes' is 'EPHESUS'.
'in' acts as a link.
(Other definitions for ephesus that I've seen before include "Ancient Greek seaport, St Paul wrote to its people" , "Comedy of Errors setting" , "Ancient Greek port in Asia minor" , "Ancient Greek seaport now in Turkey" , "Ancient Med. city" .)