As you can see, DK school dictionary says "asleep" in to fall asleep is an adverb.
Yet, according to https://dictionary.cambridge.org/dictionary/english/asleep
"Asleep" in to fall asleep is an adjective.
To the best of my knowledge, the "fall" is a linking verb which requires a predicative complement. It seems that DK school dictionary made an error. Isn't it?