... cool, showing wit - but not our one, it's beastly? (7)
I believe the answer is:
inhuman
'beastly?' is the definition.
(as opposed to human)
'cool showing wit but not our one' is the wordplay.
'cool' becomes 'in' ('in' can mean fashionable or cool).
'showing wit' becomes 'humour' (humour can mean showing wit or cleverness).
'but not' indicates named letters should be taken away.
'one' becomes 'an' (an apple is one apple).
'humour' with 'our' removed is 'hum'.
'in'+'hum'+'an'='INHUMAN'
'it's' is the link.
(Other definitions for inhuman that I've seen before include "Without compunction" , "Pitiless - barbarous" , "Beastly" , "Callously cruel" , "Cruel, bestial" .)