No, they can not legally, at least to my knowledge...
In the commercial world you can, in our OM(A) we have specific examples like incapacitation, differing qualifications for LVOs, etc.
I can log P1 time even when I’m asleep in the bunk (as I’m still the captain) but can handover command to someone else if it’s required, at which point I can't. It might be something like running out of hours before the end of the flight due to different previous duties, which I have actually had to do.
It’s harder to think of specific examples as to why you’d need to do this in a light aircraft but I suppose one might be if you had taken off on a nice day but the weather deteriorated, you had to do an IMC recovery and only one of you was instrument qualified. Might make sense to make them P1 at that point?
AFAIK there is no prohibition for apportioning P1 time amongst different people during the flight under CAA/EASA regulations. Happy to be proved wrong (with references!)