Is this correct to mean to stay lying down on your back and to ask somebody to do so?
If not, how to ask someone to keep staying lying down?
Lain is the perfect form of the verb to lie. We use the perfect when an action has been completed in the past. You are describing an action which is still continuing, so you should use the present continuous:
Stay lying down.
The post What is the Perfect, and How Should I Use It? is very useful.
"Lain" is an archaic form, and is hardly ever used in any context.
The most idiomatic ways of expressing this would probably be "don't get up", or "stay on your back", or simply "stay down".