@Lee-And-Jennifer0, if you manage a place for business contacts, not friends, a written agreement is a good idea. For friends, it’s not always necessary.
Thinking about it, I doubt that a owner would want to give full powers to a manager. Or that you really want to hold it.
How did this person enter? Let stay by the owner or by you for a while?
I suppose, if you have the keys, you could proceed like with an overstaying paying guest.
Arrive in force and start cleaning.
Have the owner call him to get him out.
get more people in (surprise arrival)
tell him he will be liable for damages, lost income and airbnb penalties.
Find a place to crash for him till after the paid stay
drop a terrarium with white mice or a snake “oh, my, where did it go? Willy! Come back! ... Did you see my rattle snake? It’s usually harmless, but must be frightened! Poor guy! Did not know it was so fast!” Best done with a fictive snake 😉
joke aside, if you do not succeed, call airbnb well in advance about the situation out of your control. They may help the guest and may waive penalties.