@Halvhjearne
That's the thing I didn't want to do, but...

So I saved all the data from my Pi, formatted the USB key, and installed a fresh version of retroPie: I was able to configure the controller.
Then I updated the system and RetroPie, tried again to configure the controller: it worked this time again.
I restored all the games now, and it's still working.
I don't know what was the problem, but now it's gone

Thank you all