I had someone rent my space and stay about an hours past their rental time. I can’t get a hold of them and now my damn profile keeps asking me to leave feedback before I can edit my site or even resolve the issue.
Peerspace sends you a email at the time the booking ends. Please choose which option is best for you. I believe you have 48 to add the extra fees.
The overtime charging for additional time isn’t the best. There has been several times I never got that email and even though I emailed peerspace because if their lack of quick replies I missed the deadline to get the money for extra hours. I even went back and forth many days later and they still never charged my client or paid me for the extra time. This of course lost them $$ since they get a cut of it on their end too. If it happened to me I’m sure it’s happened a lot and might be one reason they are losing money. My guest even messaged them telling them the same and still not over hour fee. I was highly disappointed. It should have been a 1 step email and a new charge to the cc. Sad. Wish they would make it simpler and faster to get a charge handled with less delays and stalls. Oh well they don’t care to make extra $$!! 
We make sure an attendant is able to show up at the beginning and ending of every booking to avoid problems like this. If people refuse to pay for additional hours we require them leave the premises immediately. Since we operate in this manner we have never once had an issue like the one you explained. It’s also good to show up at the end of a clients session to do a walkthrough to make sure nothing is damaged or stolen as well.