Okay, this is embarrassing.
I was doing the post-period cleaning of the little holes along the rim with a toothpick, as one does because they get crud in there, and got startled by a cat ("LOVE!" says the cat. "ARGH" says me. "YOWCH" says the divacup). So I wasn't paying attention to how much pressure was on the other end of the toothpick. Clearly, the answer should have been 'none or only a little as it passes through the hole' instead of 'approximately enough that silicone can't take it anymore', and long story short, it has a puncture now where a puncture should not be. I'm pretty sure that I have to buy a new one.
If I don't make it to the one and only store in town that carries them before next month, or have one of those wondrous "oh, you thought you were done? HA HA GUESS WHAT" moments that the menses are sometimes prone to, would it work as a temporary stopgap even though the puncture is in the region of where the blood goes? That doesn't seem very hygienic. What can I do with it besides just throw it away?