Yes, very definitely, mice play dead. So do chipmunks.
My cat is considerate when hunting, and doesn’t usually go in for the kill. Instead, she brings them to me to share. She has brought me both mice and chipmunks and lain the seemingly dead animal at my feet. Within in a few moments, the poor animal wakes up and runs under the nearest thing it can run under. Then it’s fun, fun, fun while cat and I chase after it.
I’ve learned that I have maybe a 30-second window, and I keep a box around just in case. If I can get the box over the animal before it arouses, I can easily get it out of the house.
I’ve noticed that the birds she tries to bring me are usually dead or injured. My theory is that they don’t play dead, instead they struggle, so she has to fight them harder, clamp down tighter, and they get injured.
I do try to keep her from hunting, but it’s her instinct - hard to break. And no, keeping her inside all the time is not an option. Both of our sanities would suffer.