Not necessarily to sleep/slumber, but it means to be in bed/stay in bed/get in bed, yes.
A word of warning: this expression is archaic in the extreme (Id say late 1800s, maybe?); it is not used in modern English and to do so would probably confuse most of your listeners momentarily as well as sounding very unnatural. They could probably work it out from context, though. If forced to guess I'd say it sounds more Am Eng than Br Eng too, but that is just a guess.
I wonder if the origin is related to "down" as in the direction, and thus lying down, or if it's related to "down" as in the type of feathers with which one might stuff a duvet? I'd love to find out, if anyone on here knows?
As an aside, "to be downy" or "to be a downy one" means something very different (in Br Eng at least); it means to be sharp, quickwitted, and clever or knowledgeable. It's colloquial and not much used anymore but you do still hear it sometimes.
Student or Learner