Asleep means sleeping right now. Fast asleep, fall asleep, still asleep.
It is a predicate adjective, which means it goes after the verb — the baby is asleep — and cannot go before a noun: not an asleep baby. That restriction belongs to a whole family of a- words: awake, alive, alone, afraid, ashore. Fast asleep means deeply, which is the older sense of fast meaning firmly. Note asleep at the wheel, meaning inattentive when it mattered; and my foot's asleep, meaning numb.
1adjectiveasleepSleeping right now.
“The children were asleep, so we talked quietly in the kitchen.”
asleep is a state of sleeping.
Middle English aslepe is the prefix a-, meaning in or on, plus sleep — originally on sleep. Aboard, ashore, alive and awake are all built the same way.
Middle English
aslepe
English, today
asleep
From Middle English aslepe, equivalent to a + sleep.
Etymology from Wiktionary, CC BY-SA