@Siegmar2 See also: https://service.berlin.de/dienstleistung/326217
https://www.vpmk.de/en/en/ban-on-misappropriation-in-the-capital-is-constitutional
Some short-term rentals exist with older permits that were granted when a finite number of them were made available by some districts. Others may be exempt from the terms of the law because the rental space is less than half of a shared living space (including a shared kitchen and bathroom).
Airbnb doesn't always require hosts to submit a valid permit in order to have an active listing. However, you should be aware of the fact that Airbnb submits all Berlin hosts' reservation data to the city authorities and the Finanzamt, which guarantees that there will eventually be consequences if the listing is not permitted.