The desktop app wants to auto-update to the new version. When trying to install it, the app only restarts and shows the update as available again, though. Going into settings, I could verify that I still have the old version.

This can be circumvented by manually updating through "itch-setup.exe" in "C:\Users\USERNAME\AppData\Local\itch". See post here: https://itch.io/post/15423211

So, the actual new update file is there, but for some reason, it can't be successfully installed by the app.

I'm on Windows 10, for what it's worth.