Don't engage with her in any way just yet. Speak to the head about this crazy behaviour, not so they can do anything as such, just so that they are aware. It could also pay dividends if the mad mother tries to escalate the situation and then blame you, the school will already be in the loop.
Ignoring is the best course of action initially. Just always look mildly worried about her when she does this, like you would an unsteady old lady. You're not going to get someone that immature to suddenly apologise or see reason. If you don't rise to the bait she may get bored.
However, it's equally possible she will escalate and get worse. Some people with a personality disorder do this thing where they are really really horrible to someone, pushing and pushing until the other person is horrible back. This then makes their initial bad behaviour towards the victim seem justified, they can now play martyr with others around them and it gives them excuse to be even more horrible to the victim, all while painting themselves as the true victim. It's all usually a cycle created from early trauma.
If faced with escalation from this woman, the best way to react is to put on your best Louis Theroux concerned face and ask warmly, like a grown up to a crying child, "Are you okay?" Really ramp it up. Whatever the answer, up the concern with "Are you sure you're okay. You seem to be walking into people and rambling like you've had a bump on the head. Do you need an ambulance?" Act like you've completely not noticed that she's trying to provoke you. Louis is the master at de-escalating people on the attack with excessive kindness, he's worth a watch.
Metaphorically speaking, kill this woman with excessive concern and kindness; I promise she won't know what the hell to do with that and back right off.