Hello @Susja !
Unfortunately there is no such capability in Azure Data Studio
You can achieve Timestamping the files with other methods like Powershell but to answer your question , no Data Studio does not seem to support that
I have searched the Extensions also but nothing
Submit feature requests here :
https://github.com/microsoft/azuredatastudio/issues
I hope this helps!
Kindly mark the answer as Accepted and Upvote in case it helped!
Regards