@Lucinda72 Once your original booking on Airbnb has come to an end, hosts and guests can make any payment arrangement they like. If you want to continue to stay, and the host wants to do that outside the platform, you can either agree to that, or not. If they insist on a cash arrangement, it's your choice to do that, or find another place.
Hosts asking for cash outside the platform, while you have an active Airbnb booking, is not allowed, but once those dates are over, Airbnb doesn't own the relationship between hosts and guests.
If you choose to continue to rent outside of Airbnb, I would insist on some sort of legal lease, so both you and the host are protected. If you would prefer to keep it all on Airbnb, just tell the host that- they can opt to agree or not.
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