checking the various available sources, I see lots of references to the Argentine operators reducing the power output on the TPS-43F air search radar in order to 'draw in' the RAF Vulcans to within range of the AA defences.  They certainly shut it down whenever they thought that the bombers were close enough to launch their ARMs.

On the other hand the Skyguard f/c radars remained on throughout most of the attacks, hoping to detect and track the Vulcans.  It was one of these that was destroyed by 'Black Buck 6' on 05/06/82 with a Shrike ARM.  At least one source suggests that the missile was already in its terminal dive when the Argentine crew apparently switched off the radar.

NB.  The earlier 'Black Buck 5' mission only inflicted minor damage on the TPS-43F radar as the ARM impacted 30-45 feet away before exploding.  Had it achieved a direct hit then the radar might have been completely destroyed - potentially changing the air war over the Falklands completely.

A good source is 'Falklands Air War' by Chris Hobson (ISBN 1-85780-126-1).  It can get some getting used to as it catalogues the war day by day - so searching for individual BB missions is time consuming if you don't have the dates already to hand.