I agree, the girl sounds lonely. Annoying yes, but still. Her mom can't do anything about it when she's at school.
This really is a school issue, one that should be brought to the attention of the teacher. Perhaps the teacher can help the girl find new friends, put her in groups with other children. Your daughter can still be her friend and introduce her to other children so they can all play together. Her mom can create playdates for her daughter with other kids or put her in after school activities to help her daughter branch out.
It's hard and confusing for a child to be best friends with another, then she doesn't want you around anymore.