Notices banker's out of line and arrogant (8)
I believe the answer is:
cocksure
'arrogant' is the definition.
(synonyms)
'notices banker's out' is the wordplay.
'notices' becomes 'cocks' (I can't explain this - if you can you should give a lot more credence to this answer).
'banker's' becomes 'Ure's' (River Ure, something with banks).
'out' means to remove the last letter.
'ures' with its last letter removed is 'ure'.
'cocks'+'ure'='COCKSURE'
'of line and' acts as a link.
This may not be correct. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for cocksure that I've seen before include "Use crock to be confidently certain" , "Annoyingly self-confident" , "Foolishly over-confident" , "Offensively confident" , "Very confident like male fowl" .)