Alarm caused by prince, not British (5)
I believe the answer is:
alert
'alarm' is the definition.
('alert' can be a synonym of 'alarm')
'prince not british' is the wordplay.
'prince' becomes 'albert' (I've seen this before).
'not' indicates named letters should be taken away.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'albert' with 'b' taken out is 'ALERT'.
'caused by' is the link.
(Other definitions for alert that I've seen before include "Watchful; warning" , "Watchful, sharp" , "Watchfully awake" , "sharp-witted" , "Vigilant, watchful" .)