I have just read an article about the word "asleep". It says that there's no agreement among linguists whether "asleep" is only an adjective or can be both an adjective and an adverb. Dictionaries also define this word differently, for example:
asleep - The Oxford English Dictionary - Adv., pr. Adj.; Webster's New Collegiate Dictionary - Adj.
The definition of "asleep" as an adjective is the following:
1: being in a state of sleep 2: dead 3: lacking sensation: numb 4 a: inactive, dormant b: not alert: indifferent
And here's the definition of it as an adverb:
1: into a state of sleep 2: into the sleep of death 3: into a state of inactivity, sluggishness, or indifference
Still, the point of view that "asleep" can be an adverb is not common.