I was told once that there is a theory consisting of just a pairing function that is stable, although I cannot find a reference for it. This motivated my question, which is essentially the title, although really a simpler question should be asked first:

Is there an expansion of $(\mathbb{N},<)$ by a pairing function that is still nice in the sense of neo-stability?

Obviously it can't still be o-minimal or dp-minimal, but in principle it could be NIP. Assuming a positive answer to that question it's natural to ask how much more structure is compatible with 'niceness,' hence the question in the title:

Is there an expansion of $(\mathbb{N},+,<)$ by a pairing function that is still NIP?

I'm primarily interested in the theory of the expansion being NIP, rather than the weaker notion of the structure itself being NIP, as introduced by Khanaki and Pillay, although I would be interested in that if the harder question has a negative answer.