Yes, but not by anything even vaguely resembling credit cards as we know them today. The whole system is insecure, inefficient and unreliable by design, and the credit aspect introduces an extra level of legal and financial mess, particularly for those people who just want a convenient payment method but don't actually need to defer settlement. Card payments may be the perfect example of the dangers of monopolies and oligopolies.

I'm guessing their Network agreements with Visa and MasterCard would stop them. You could issue a stripe card that only works with stripe merchants. But I can't imagine visa allowing their imprint on a card that can both process anywhere AND be used to slowly cut visa out of the picture.