A couple of comments. (I should note that I'm not a parent, but that I work with kids at Church.(
You ask about punishment. But you haven’t said why she lied. Do you know? I’d start out by trying to understand what happened and why before considering punishment.
Assuming she actually did lie (schools do make mistakes), the Sunday School kids I teach seem to lie primarily because they’re afraid. They don’t want to upset their parents, or they’re afraid of being punished. Maybe punishing them for lying will help, but maybe it will make them more afraid and more likely to lie in the future.
The school has already done something. You’ve already done something. None of the psychological evidence I know suggests that more or more severe punishment is going to help. What matters is to hold people accountable for their actions (and in fact there are ways to do that other than punishment). At this point I’d suggest trying to understand what happened, and see what you and your daughter working together can do to prevent it in the future.
