The Pyra is a general purpose computer, and those predate the internet and arpanet. If you've already downloaded openstreet map files for your location, you can still use maps, and you can still run any games and emulators you've already got. Now you do tend to get those files via the internet in the first place, and if the internet suddenly stopped existing before the pyra came out, we'd probably end up posting around sd cards like PD houses used to mail out floppy discs to people.
It's not really like your phone, which would pretty much revert to being a dumb phone without a package interface to the internet. And without the internet, openstreetmap itself would probably die, but the prexisting data wouldn't suddenly stop existing.