@Maria924
Hi Maria - you dont say which property, but two of your listing have Instant Book switched on, so certain types of guests will be able to book without waiting for you to confirm. I assume it was one of these.
The guest is correct that you need to either honour the booking (as the calendar was open and IB switched on) or cancel for them.
If the guest contacts AirBnB and gives their side of the story you may find you are penalised for this error - you would be better off contacting them and explaining the misunderstanding and hope they will not penalise you, and also of course refund the guest the full amount.
Easiest way to contact AirBnB is via Twitter - search for AirBnB help. Or do a search on this site for Contacting AirBnB - there are lots of guides on how to do so.
Finally you say there is no record of their reservation: go to your Calendar and look at the dates they have booked. Click on the booking and it will open the details. Or go to 'Your Reservations' and find it.
What I do not understand however is your comment that you have received payment for the reservation - AirBnB do not start to process your payment until about 24 hours after your guest has checked in.