My DDs are younger than yours but maybe if it the first time her DC is coming to play and she doesn't know you well she just wants to meet you and know you a bit better....presumably you weren't hoping to have to amuse an adult all day too so a bit of a pain as you won't be able to get anything done...
At 9??? No no no!!! Why on earth do they feel the need to do this unless it is a hell of a journey to and from your house and they don't drive. DS is 9 and he plays out in the small estate I live on with lots of other kids. They are always in and out of each others houses and I certainly don't accompany him - I would be very odd if I did. And it gives me more time to MN erm clean the house
If I didn't know the parent or child very well I would always make the first play date for a short coffee, so I could snoop around get to know them a bit better first. Then I'd be comfortable to drop and run in the future. My DS is only 5 though, so I probably wouldn't take him somewhere for the entire day unless we knew them very well already. So, I think YANBU but also I think the other mum isn't either! Sorry, maybe a shorter date for first-timers in the future?
At 9 I would drop and leave/expect them to drop and leave - if it was a first playdate then maybe a coffee but no more. If it was a really good friend of mine and dc's then maybe lunch etc but that would be clear from the start and an exception.
I think it may be because of how it came about that her DD is coming to yours. She rang you? to say her DD would like to play with yours? Presumably she expected to have your DD? You then invited her DD round for day, give them lunch etc, she may have just thought you were inviting them.
When she arrives you can simply say, come in and have a coffee, you can then go if you want, get some things done and come back at xpm.
have you answered the text ? Just tell her she is fine to leave her DD with you until she gets back from her appointment - no need to hang around until then. Or suggest you could drop her home a bit later in the day.
Maybe mention you work from home so are hoping the kids will occupy themselves for at least some of the time.
Why would anyone stay with a 9 year old, in teh normal run of things? No need.
Maybe she has misunderstood you. This appointment is the perfect opportunity. You should say, 'oh feel free to head off after lunch, have a bit of time to yourself. Of course your dd can stay while you are at appt. Just pick her up when you've finished.'
You need to be totally firm - say (rather than text so there is no misunderstanding) - leave DD with us at 10am (or whatever), I will give her lunch and you can pick her up after your appointment. Make it absolutely clear.