Ha, I've found at least a quick-and-dirty solution, see image. I turned the Show etc switch off, now red in the image.
The problem may have its roots in the fact that I used to have the paid Avast Security Suite (or whatever it was called) but switched back to the free version of Antivirus when the licence expired, but renewed the paid SecureLine. Maybe the ghost lingers on from the paid Security Suite version.
Raeken, thanks for the pointers! I didn't come to think about the possible connection between the paid and free versions until your comment.
If you have any hints as to how to remove the SecureLine component from the now free, previously paid-for Antivirus, I'd be grateful. Otherwise, I'll just leave things as they are now. At least I got rid of the annoying superfluous taskbar icon.