I know my problem looks really crazy. I tested a HTTPClient sketch using HTTPS successfully.
After soldering the headers I repeated the test and it fails (Wifi connection disconnected during the GET call). Using HTTP the test succeeds.
Is it possible that soldering (probably my soldering iron was not grounded) causes such a specific problem ? Everything else seems to work.
I repeated the same procedure on a second Wifi Kit 32 device with the same result.
Do you have an idea what happened ?