OpenStreetMap doesn't have that. Services downstream from OpenStreetMap, like Komoot, OsmAnd, and Organic Maps do, but OpenStreetMap doesn't have any offline mode or anything (and only has very rudimentary GPS tracking on the website, for centering the map on your general location to make editing easier)

I see OpenStreetMap primarily as the dataset and community, not the website. So OsmAnd is in that same ecosystem for me and qualifies as "OpenStreetMap" in a wider sense of the term. For me, at least.