I've been doing a bit of testing of some of the offline functionality in my app(s). This generally means, loading the app, and some data, then putting the device in flight mode to see if things didn't break.
First of all, I found that sometimes PowerApps still tries to make online calls, even though it's not connected. But after navigating to a new screen, sometimes the 'loading dots' appear in the top of the screen, and after a while an error notification is shown. I'm pretty certain it's not my own code/formulas doing this, as I designed these screens to work with cached local data. (The error message is typically something like "Not a valid connector Error response".)
Second (and this is way worse!), I discovered that it's possible for 'something' to become corrupted in the app if I'm offline, and navigating to a new screen. Whenever this happens, the screen becomes just a mess of controls/components in their initial state, and the app doesn't respond any more. I can't do anything else other than restart the app. However, in addition to having to restart, this also seems to permanently corrupt that screen -- so even when I disable flight mode, the screen is still broken when I navigate to it. And the screen remains broken until I empty the PowerApps cache.
While the last problem does not occur in all screens, there are some screens that consistently display this problem whenever I navigate to that screen while I'm offline AND the screen has not been opened before.
Not sure how to approach this. Can provide a video demonstrating this problem, if necessary.
/edit: tested this both in the iOS app and the Android app. Problem is present in both.