Thank your for your replies and the pictures!
I already found some photos of fish with Oodinium, too – and I am quite sure that my fish do nhot have Oodinium but, as Peter assumed, some heavy funghi infestation!: No white spots but the mentioned white, fur-like cover that absolutely looks like funghi to me.
So on the one hand, I’m relieved now to not have Oodinium in my tank. On the other hand, I still have this very acute funghi problem – and I am really afraid of the sight that is awaiting me at home.
With kind regards and fingers crossed,
Michael