"Ní bean mé!"
Translation:I am not a woman!
I'll have to disagree with both Postillion and RyanOkushi here: It does not necessarily mean the same thing; and, it is grammatically correct.
There is a subtle difference between saying "I am not an X" and "I am no X." The former is a purely factual statement, while the latter expresses that one does not have the character of an X. Saying "I am not an author" simply expresses that one's profession is not writing. Saying "I am no author" would almost imply that one lacks the skills or characteristics associated with being an author.
Sometimes, "I am no X" can mean the same as "I am not an X," but in those cases, the former sounds rather old-fashioned. An example would be Éowyn's famous line from Lord of the Rings, "I am no man."
TL/DR: This sentence is a copula usage, therefore "Níl" (a form of "bí") cannot be used for the same reason as "Tá" cannot be used in the sentence "Is bean mé".
The difference is that "Ní" is the negative form of the copula (previously seen in its positive form "Is") while "Níl" is the contracted form of "Ní fhuil".
What exactly is "Ní fhuil" you may ask? Well, "Ní" is the preverbal particle of negation (meaning "not"), basically, you put it in front of a verb to make a positive statement negative. The "fhuil" ("fuil" but lenited because of "Ní") part is actually a form of the verb "bí" called the 'dependent' form. (If you remember, "Tá" is the present tense form of "bí".) It is used when the verb doesn't stand alone (in this case, because of the presence of the "Ní").
An example may help... If you took the positive statement "Tá mé mór" (I am big) and wanted to make it negative, and 'see' it step by step, it would look like this:
"Tá mé mór" --> X fuil mé mór --> Ní fhuil mé mór --> Nil mé mór.
*These forms are not really valid, they are for illustrative purposes only.