I suppose we could argue that "asleep" describes how she lay in the bed, but it seems more logical to think of it as an adjective modifying "she". It should be clear if we rephrase:
She lay on the bed, asleep.
Compare this to something like:
She lay restlessly on the bed, unable to sleep.
Where "restlessly" is an adverb describing how she lay. We could alternately phrase this as an adjective:
She lay restless on the bed, unable to sleep.
My impression is that "asleep" is only an adverb when used as part of the phrasal verb "fall asleep".
After marching all night, the soldiers stumbled into the barracks and, almost immediately, fell asleep on their bunks.
Otherwise it seems always to be an adjective.