I shan't tell her, I promise. A natural refusal, and it means the same as I shall not tell her.
Shan't is the short form of shall not, so I shan't be late tonight means the same as I shall not be late tonight.
You will hear it far more in the UK. In the US it may sound a bit old-fashioned, and an American is more likely to say will not in the same spot.
We shan't stay long. A natural refusal, the same as we shall not stay long.