Still overjoyed after giving up The City (6)
I believe the answer is:
static
'still' is the definition.
('static' can be a synonym of 'still')
'overjoyed after giving up the city' is the wordplay.
'overjoyed' becomes 'ecstatic' (I've seen this before).
'after giving up' indicates named letters should be taken away.
'the city' becomes 'ec' (the City of London postcode).
'ecstatic' with 'ec' removed is 'STATIC'.
(Other definitions for static that I've seen before include "Still - radio interference" , "Inert, motionless" , "problem with radio reception" , "disturbing thing" , "irritation for the listener?" .)