Well, to make things more interesting. Only yesterday-evening I powered down the phone to make sure I could test again today with a clean state. And today the signal icon gives _no signal_ at work the whole day, however most (if not all) apps do have Internet access. But I cannot call out, instead I get the message: "Mobile network not available."
3G or HSDPA (I don't know what it is using right now) works flawlessly ! So apparently you don't need a Mobile network for having 3G :) They should call it Networkless WAN :-D
Switching Airplain mode ON and OFF solved this weird situation.