I have just started having issues in ArcGIS Pro 3.4.0 with the basemap not loading. I see all the various basemaps in the list but no matter which one I pick I get a warning and nothing loads
It would be worth trying a clean uninstall and reinstall of ArcGIS Pro
I've seen this fix many similar issues in the past
This post was never marked as solved, and it does mention a recent Pro update or even the version being used, but since the timing of the posts are fairly close, I thought it would be worth mentioning you're not the only one to experience a problem like this recently.