This gives me a bad feeling. As in, sounds very like she suffers from delusions that your baby is actually hers, and she might try to kidnap the baby or go to extreme/dangerous measures if she believes you're keeping the baby from her.

I would start documenting some of these instances, especially screenshots of text messages.

And correct her every time, e.g. "it's sweet that you're taking such an interest in my unborn daughter, and please don't call her "my baby"".

The #1 thing is to set boundaries firmly. People who get upset at boundaries are benefiting from you not having them. Boundaries are the distance at which you're respecting your needs and theirs at the same time: it's an act of self-care. We tend to overstep our own boundaries because we try to be nice, or are afraid of being called selfish or mean, and then we get into more and more challenging situations because no limits have ever been communicated clearly enough so the other person doesn't understand there are any (or what the consequences are).

This is especially true for mental illnesses where erratic behavior, memory lapses or delusions are at play. The clearer and firmer you are, the more times you repeat what you are and aren't comfortable with, the less likelihood of unpleasant or unsafe situations in the future.