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You can use Open Street Mmap and a suitable app of your choice. OSM is strong on the local level. Check out https://organicmaps.app/

Does this show on the car's display or is it limited to showing on the phone's screen?

Organic Maps supports Carplay, but not yet the Android variant You can however not use both the phone and Carplay at the same time

If you need that, you can try https://www.magicearth.com