OK, this is strange.
I killed the pod, waited until the server came up again, and tried to “mvn deploy” a file again.
And now it seems to work 
Before that I also added a task to repair any maven metadata on the target repository.
Sidenote again: I run the Nexus 3.29 version.
So, sorry for the post. No idea why it works now but I guess there is/was an issue with the internal scheduler from Nexus.