Is it obligatory to continue paying the monthly sum to the poor woman, after the death of the mother who vowed that and promised the poor woman it would continue for life?

What your mother promised that poor woman is not an explicit will, for a promise to bequeath is not a will. You are not obliged to give anything from the inheritance to that woman as long as the matter is uncertain whether it is a will or not, and she does not deserve anything merely based on doubt.