This is not uncommon. Often, when this does begin to happen, parents get worried and might let their child stay home with them, thinking their child is just having an off day. Then, the child does it more and more. This is nothing they plan to do, but after having a parent keep them home a time or two, it's almost like a confirmation that their resistance to separate from their parent is well founded.
My little sister went through this in elementary school. She would always get upset when I would drop her off and at the end of the day she was always fine. She finally told me that there were a few kids teasing her because of her name, Monica. It was around the time of a certain presidential scandal and the kids started calling her Monica Lewinsky. When she finally got up the nerve to talk to me about it, we role-played ways she could shrug it off as if the other kids were silly and boring. That totally worked.
You may have asked your daughter what is making her want to stay with you. If there doesn't seem to be a reason like a classmate that teases her or that she is suddenly worried about her too curly, too straight, too long or too short hair(or something that seems completely strange to an adult), then it may work for you to try a fun positive reinforcement tool:
It may help to start doing a "pick-up surprise"--nothing big and it would only be for when you're picking her up. You can hide a special surprise--stickers, temporary tatoo, pretty hair clips, cool plastic bugs--whatever small thing(preferably non-food) she likes that you can easily find in bulk at the drug store or dollar store. Next time she is having a rough time at drop-off, just tell her, confidently and in a relaxed way, "Guess what...When I pick you up, I'm going to have a little surprise for you!" Any resistance to that(or even, "No, I don't want a surprise! I want to stay with you, Mommy.") should be met with a short, loving, confident reiteration. Then, when she comes back you can leave a surprise on the seat or even make a game of having her find it in your car. That should help relieve the issue within about 2 weeks, if there is not a separate reason for her reluctance to go to activities, like there was in my sister's case.
Good luck! :)