I would just tell them you are sorry but as AP shows it has been delivered there is nothing you can do however they should contact AP again and fill in a complaint form if they did not get the parcel. Deliveries have all sorts of gps tracking, photos etc now I believe so that may shed some light on what has happened.

You should be covered in the event of an ebay or Paypal claim although it does not mean for sure that the buyer is telling porkies, a delivery person may have stuffed up, it may have been safe dropped and somebody has pinched it.