@Kristīne0The guest should be subject to your cancellation policy, and as this was a same-day cancellation, they shouldn't be entitled to any refund. However, if the guest calls Airbnb and makes a complaint, the agents can override your policy and decide to refund the guest. I would hope they wouldn't, but you never know: again, Airbnb sometimes favours the guest.
Bear in mind that the guest can still review you because they cancelled on the check in day. Offering a refund may or may not save you from a bad review, but just keep it in mind.
As they've cancelled, I wouldn't bother calling Airbnb at this point.