In Latin and Greek, when a negator appears in an absolute construction (ablative absolute, genitive absolute), it is generally taken to negate the predicate within that construction:
hostibus impetum facientibus "the enemy attacking"
hostibus non impetum facientibus "the enemy not attacking"
Do we ever get instances of a negator applying not to the predicate, but to the whole absolute construction? A made-up example of what I mean:
Caesar se recepit non hostibus impetum facientibus, sed fessis suis. "Caesar retreated not because the enemy were attacking, but because his men were tired."
Here non is semantically outside the absolute construction rather than applying to a constituent within it.
The same question could be asked of Greek genitive absolutes; in fact this question occurred to me reading an answer by Draconis to a Greek question, in which he translates οὐ παραμιγνυμένου αὐτῷ κασσιτέρου as "not because it's mixed with tin" (οὐ negating the whole construction) rather than "because it's not mixed with tin" (οὐ negating only the predicate).
My sense is that for both Latin and Greek, this kind of usage is either nonexistent or very rare, and that meanings like these would be expressed with some other construction (e.g. non quod, οὐχ ὅτι). Thoughts?
ETA: prodded by the examples in Mitomino's answer, I'm now thinking this is really a question about information structure rather than negation. Maybe what's odd about my made-up example above is not that it's negated, but that it's acting as the focus of the sentence. In that case the question can be recast as "Can an absolute construction be focused?" Answers to this question would be equally welcome.