Caregiver refuses to stop smoking after being politely asked repeatedly. Any advice?

You have got to be kidding me. Is this a real question? The answer is obvious. If the caregiver works for an agency, get a new caregiver. If the caregiver works for herself, find a new caregiver and then kick the smoking caregiver's butt(s) out the door.

NOTE: It is very easy to tell another person what to do. It's another thing to do it. As an example, we had a small group of caregivers take care of my mother until she died, probably two years. Sometimes I would come into the den where my mother was lying in her chair to find one of the caregivers cracking and eating nuts on my mother's midsection. She did bother to put some newspaper down. Why didn't I just kick her patootie out the door? I don't know. I probably will never figure it out. As it turned out, the woman who was the ostensible owner of this little agency stole money and, of all things, eggs. At one point, we escorted the whole bunch out the door and hired an agency. The agency is not a gift from heaven but at least they aren't stealing eggs.

That is a decision, I would think, for your friend herself to make. I am afraid not an easy decision to make. We are in a time where now there are not enough workers for jobs. Cheeseshop closing on my favorite shopping street "can't get workers" and the B of A bank had THREE of its banks closed last week; not enough personnel. Your friend will have to weigh whether they can come to an agreement of when and where her worker will smoke, or not, providing she can tolerate smokey clothing. The 2nd hand smoke is, of course, not good for her. That can be recognized by all I would think.
I wish her luck in her decision .

Its not so much that the friend suffers from Asthma, it the gaul of the Caregiver. Even if the friend didn't have a health problem she has a right to ask someone to not smoke in her house. If this CG is from an agency, report her and ask for a replacement. If not, start looking for a new one and when friend has found one, fire the other.

At 72 I still cannot believe the entitlement of some people.

That's a no brainer! Of course she should find a different caregiver. The fact that this caregiver(and I use that term loosely, as it doesn't appear that this person really cares, because if they did they wouldn't smoke in front of your friend with asthma)shows no respect for their client speaks volumes about this person. Time to part company. But not before reporting this behavior to the agency they came from.

Fire her.

Is this an agency caregiver?

YES.
See if your friend needs help with this as far as calling the agency or whoever is sending them out to care for your friend.