Can a Pressero store process payments using P-Cards?
Probably, as they can generally be used like any other debit/credit card depending on who issued the card and what restrictions they may have set.
P-Cards can be used wherever the issuer of the card (Visa, etc.) allows it to be used. So it's more of a question for the issuer of the card, whether they allow the card holder to make purchases at specific merchants matching the merchant category codes they allow/restrict.
For example, if Visa is the issuer of the card, and your Pressero store accepts Visa payments (through Authorize.net, Paypal, etc), then you should be able to accept Visa P-Cards provided the merchant they buy from (you) falls within the allowed categories controlled by the account owner (your corporate customer that issued the P-Card to its employee making a purchase at your site).
If your customer is having issues making purchases using a P-Card, first check with the issuer to see if any restrictions are in place. Otherwise they can be used like any other debit/credit card.