She'll still be needing naps etc won't she.
Compromise, she can go out with them after her nap and home for tea, bath and bed.
I wouldn't want my DS away from me all day, I have to put up with it now cos he's 5yo, but at 9.5m? No. It's OK not to want to have her away from you for a whole day. When I was in egypt, women H knew, relatives, colleagues, friends etc were forever trying to carry DS, and I hated it. They thought they were helping. They weren't.
There is always the desire for GP/Family to want to do things with the DC now, buy things now, but there is plenty of time.
Tell them not to worry, that 9m is a little too young to go off on her own all day, when she still needs milk, nap, is not fully weaned etc.
The next trip she'll be 2yrs + and will be able to keep up, and not need so much mummy attention, and you will be better prepared for it.
As you are staying with them, as someone said, they will have the time with you all and won't need to panic and take her out.
Agree with the comment about taking the emotion out of it. Just calmly say, no, that won't work on many levels, it's too early for a whole day, when she's woken from her lunchtime nap, fine, and let them take her for a couple of hours.
You are right. You are her mother and if it's not OK with you, it's not OK.
If you were happy about it, you'd make it work, but YOU are not ready to do this. It's fine, I don't blame you at all and would do the exact same thing.