AM in general, but yes. A good example would be the immediate aftermath of Hurricane Katrina - local New Orleans AM stations (the ones that remained on the air anyway) were the only source of information for many people as everything else had been knocked out.
IIRC, WWL AM 870 remained on the air.
Yes, they did. 24/7 for weeks until the other stations could start to get back on the air. That was up there with Murrow and Cronkite relaying the Blitz live from London via SW in 1940.
WWL beams its 50kw northward and usually puts in a strong signal here at night.
First reports on early Monday morning following hurricane landfall seemed optimistic and indicated that the city of New Orleans avoided
a direct hit. Then the levees started failing...