I was travelling in Germany and rented an apartment for a few days. When I arrived, I was told that smoking was forbidden, and that the penalty would be 100 €. I've never smoked and so I had no problem with this. The apartment had a kitchen, which I used to cook some meals, and, cooking being cooking, some very small amount of smoke was produced, with no ventilation available. The day I departed, I received a bill for 100 €, claiming that I had broken their rule and smoked inside the apartment. I attempted to contact the owner by email to discuss the situation, but received no reply. It seems to me that the owner is trying to extort extra money out of me, and I don't have any clear way to defend myself. Is there any course of action available to me?
If you haven't paid this bogus fine, do nothing.
They've accused you of something you didn't do, and you've refuted that claim in writing. That's all you need to do for now. Since it's via email, you've got copies. You can print them if you like, so that you've got a hard copy.
If they do later respond, refuse any communication via phone even if they call you directly - just say that you'd rather communicate via email so that there's a record of everything said. Keeping written records is really important. It may also be a good idea to check review websites to see if anyone else has had similar complaints.
Chances are they won't bother taking any kind of legal action to recover the 100€ fine because, even if it were legitimate, it would cost them more in time and money to recover it. Even less likely if, like you say, it's a false claim.