@Ana1971 If I had a really great guest who just wanted to extend their stay by a few days or a week, I might very well do a private deal, just as I let great guests who've stayed with me know that if they ever want to come back, they can just email me directly. Many hosts have their own websites they rent through, as well as using Airbnb and other booking platforms. Just because a guest first came to me through Airbnb doesn't mean Airbnb owns the relationship between me and that guest forever, once their Airbnb booking has come to an end.
But I'd never do such a thing with a guest like you describe, and I've reported every guest who has contacted me through Airbnb and sneakily tried to get me to negotiate with them directly.