Former preschool teacher here, and now a mother-
i'm seeing people say she might just not be ready, instead of just pulling her why don't you set a date in your mind- maybe Jan 1st. give it that long and then know that if it doesn't work out then at least you gave it a good shot. That might make you relax a little bit about it, knowing It isn't a matter of her winning (which i think it would be if you stopped right now) but that you won't have to deal with this forever.
Kudos to the teacher suggeting the playdate, i feel your hesitancy on this, and as a shy person, the thought of playdates with people i don't know at all freaks me out, BUT i do think it would really help your daughter, Ask the teacher to suggest someone that she things your daughter might bond with, and send a note for the teacher to give to the other mom, with your contact info and just explain that your daughter is having a rough time adjusting and would this other child and her caregiver consider coming to your house for just an hour so your daughter could see that the other children in the class are kind and fun to be with. YOu could prented to have a tea party or something and get your daughter excited about being sortof a party instead of some kid coming to touch all her toys. I would make sure the playdate is at your house on your daughters turf, and keep it short and hour would be plenty, and most likely the other mom would want to stay, that ensures you only have to make consversation for a short time. Serve a snack and have some coloring and a few toys out and see how it goes. You might need to try a few kids to see which family you click with but this would be the single best thing to help your daughter out. don't forget the neighbor boy too, as long as he is a gentle kid. Boys and girls this age don't care at all about gender when they play.
Another helpful thing would be to role play "school" at home, let your daughter be the teacher and welcome you to school and go through the day's schedule with her or let her pick what she likes best about the day to "play " with you. She could read you a story (one she knows the plot to and can retell) you could plan out a little snack and have her pass out the napkins, you could have "circle time" and pretend to sing the days of the week, get out the playdough or puzzles that sort of thing. Ask the teacher to write out the schedule and talk to your daughter about it so she knows what comes next .
I've had kids before that missed mom all day and would just sit and not participate, but they were always watching and i think learned quite a bit even from their spot off to the side. Some times other kids inviting them to play helped, If she is wailing all day though and continues into the month of Dec then i would pull her and try again next year. It might help you if there is an observation window to watch and see how bad it really is an hour into class, two hours etc