No end of tough argument in school (6)
I believe the answer is:
harrow
'school' is the definition.
(famous UK public school)
'no end of tough argument' is the wordplay.
'no end of' means to remove the last letter.
'tough' becomes 'hard' (associated in meaning).
'argument' becomes 'row' (to row is to argue).
'hard' with its last letter taken off is 'har'.
'har'+'row'='HARROW'
'in' is the link.
(Other definitions for harrow that I've seen before include "Soil cultivator - English public school" , "worry" , "Distress greatly with agricultural soil-breaker?" , "Cultivator that breaks up the soil" , "Break up (ground) - distress (feelings)" .)