Demanding slice off European (6)
I believe the answer is:
severe
'demanding' is the definition.
(severe can mean strict or demanding)
'slice off european' is the wordplay.
'slice off' becomes 'sever' (I've seen this in another clue).
'european' becomes 'E' (abbreviation e.g. EU).
'sever'+'e'='SEVERE'
(Other definitions for severe that I've seen before include "Extremely difficult to endure" , "Harsh - extreme" , "Critical" , "Drastic" , "Intense; harsh" .)