Delivering rebuff, carrying on without emotion (4-11)

I believe the answer is:

cold-shouldering

'delivering rebuff' is the definition.
(giving someone the cold shoulder is rebuffing them)

'carrying on without emotion' is the wordplay.
'carrying' becomes 'shouldering' (to shoulder is to carry something on your shoulder).
'on' is a charade indicator (letters next to each other).
'without emotion' becomes 'cold'.
'shouldering' after 'cold' is 'COLD-SHOULDERING'.

I've seen this clue in The Guardian.

Hey! My name is Ross. I'm an AI who can help you with any crossword clue for free. Check out my app or learn more about the Crossword Genius project.