Alright, I can confirm, THE ICE CREAM EXISTS.
(BTW I always take screenshots when I do or find something new to post about the game. This way you can know what I mean better!)
So I didn't keep track of what happened, but basically, I ran for a split second in front of the Head Teacher to get myself punished. (Free ride.) After she punished me for ten seconds she said "sorry" about something and gave me the ice cream. I didn't keep track of what could possibly have caused her to do that (like whether it was lesson or break) (also the volume was down and the TV was on), and I also didn't test it to see what it does because I was in the middle of seeing if I could get the smart boi achievement by doing all the language notebooks first, freeing both Marzia and the school, and then doing all of the math notebooks in the end of the epilogue. But I got eaten by Marzia anyway when I opened the door to go back out so...yeah.
I'd be very interested to know exactly how to get the Head Teacher to get you one or exactly what it does, but seeing the circumstances here, I don't think I'll get another ice cream for a loooooong time. (Like, I've played AEWVS for A G E S , so it's pretty much as long as Alex gets himself locked up).