"I'm not a woman." should be accepted
Of course. If it hasn't been, then it must have been some bug.
The "correct" answer was "I am not a woman" I put "No I am not a woman" and it was still wrong. Please clarify how is my answer different from the actual answer.
You would need a comma in your English sentence for it to make sense and the translation would be different:
No, I am not a woman = Nie, nie jestem kobietą
"I am not a woman" and "No I am not a woman" are two different sentences.