Yes, but it depends on the app.

I run both STRAVA and RideWithGPS at times.

Strava seems to be much better at learning local routes and downloading maps. But, if you get in a new place, then it doesn't download the maps.
RideWithGPS will still work without a phone signal, but it seems to forget where you've been, so maps can be quite sketchy.

The ride data, of course is still saved for later download and analysis.

I think there was a big thing a while ago about Google Maps that allowed you to configure a local map cache. Perhaps that would also help apps that utilize the Google maps.

I know STRAVA will allow you to save rides, and then queue them until it gets internet access. I think RideWithGPS also does that.

Many applications also let you run the application without a SIM card, and only transfer data when you get a good WIFI wireless connection.