Could it imply that she fell asleep in her chair with her head on the table? That's the intended meaning. Or does it strike you as she was lying on the table?
N2g, as a non-native speaker, the reading of the sentence gave me the impression that she was sleeping, lying on the table. But as Optilang said, it might not seem so to the native speakers.
To tell you the truth, gentlemen, It never occurred to me that she was anything other than prone on the table. It seemed like GG got the same impression.
In my experience a native speaker would say "fell asleep with her head on the table." "Fell asleep on the floor/bed/table/couch/porch means what it says." "Fell asleep at the table would be assumed as "in a chair." "With her