Problem solved. The antivirus software I use was blocking the access of Keynote to the folder where it stores a.o. automatically saved files.
The antivirus now came with a popup window:
'Activity blocked. Keynote tried to gain access to the folder "Library", but has been blocked according to the policy of DeepGuard."
The window, however, offered the option to allow access. Which worked. Also, the history of the blocking was logged in the antivirus app.
The antivirus software is called 'KPN Veilig Virusscanner' and is coming from F-Secure. The current version is 19.5.