The baby has been down for 3 hours.
I found this expression in the language teaching book and it says the meaning is sleeping, but I think this is not a popular expression. Is it?
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In this case “down” most likely means “lying in a relatively low position, like in a bed, in a crib, even in the floor.
Down has many meanings, like “depressed” (but babies are usually not depressed), “destroyed” (there was a gunfight until the shooter was down), it might mean “on the ground floor” and many other meanings.
The “ghost busters” joke: “putting someone down” also means “insulting them”.
Talking about putting a child "down" does indeed mean that the adult has forced/coerced the child to sleep.
Is this common? Yes, very, at least where I am (west coast USA).
If this doesn't sound common to you, it is probably because this use of the word "down" is used almost exclusively for children. This can also be used with adults, but it has a very different connotation, in which the connection with sleep is used as a euphemism for unconsciousness, ie "the fighter was down for the count."