What to do

It’s your child. So no smoking where your child is. That’s super bad for your baby. Can she not adjust to that? Your child shouldn’t be there.
I wouldn’t make any compromises about that.

She says she smokes under the extractor hood, like that helps. You can't really talk sense to her, and my boyfriend and I end up arguing about it, which I don't want. It's like that now; she smokes all day in the house, and when I come home, she goes under the extractor hood, but it really doesn't help.

Tricky, it's her house. But it's also your child. If you don't want that to go up in smoke, you'll have to stand your ground and make it clear that you really don't want that. Given the risks of secondhand smoke, I totally agree with you. And I can completely understand your cautious feelings; I also lost my first child and am very protective of my second little miracle. For babysitting, I'd come up with something else. My mother-in-law also smokes and wants to babysit, but she already started cutting back before our little one was born because quitting hasn't worked out for her yet, and she no longer smokes inside her house. She now smokes outside and doesn't want anyone to smoke inside anymore. I can live with that.