The light can be turned off completely. Definitely.

Hier der Originaltext der Bedienungsanleitung:
You can switch off the display completely by pressing the display button, in
case you do not want any light - including time indication - coming from the
Sleep & Wake-Up Light.
Normally, if you do not touch the display for some time in any menu, the
display reverts to time and status indication only. When the display is off, no
time indication is shown.

Yes, the clock is always lit but it is not so bright. You can't turn off the illumination of the clock.