I use a Garmin Monterra on my bike that operates under Android (old outdated version though).

The Garmin apps themselves are crap but other apps like Osmand, OSMtracker and others are working fine.

When I posed my question to the list, I was thinking of a dedicated
GPS/satnav device, not an app on a smartphone or some other Android device.
I am a cycling enthusiast, and I don't want to use my smartphone while
riding, so I prefer a dedicated GPS device. I suppose such a thing that
supports open mapping format does not exist (yet).

Mark Bradley