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On Freeview, yes - it's just an extra channel or two, though admittedly with no EPG entry so recording has to be manually scheduled. On Sky, no.
I've recorded BBC1 red-button programmes before now. It's a bit hit and miss, though. They seem mostly to be on Freeview channel 301, and they will stream for a while and then stop.
A bit of a faff but would it not be possible to add the red button channels to the channel list and then set up a manual recording?