The real 'Batmen' served during World War II
In 1942 the California State Guard was trying to protect the state from Japanese invasion and organized a unit of "bat-men" paratroopers.
The idea started in an Aug. 1941 issue of Mechanix Illustrated, a now-defunct magazine that ran until 2001. Former Army Maj. Malcolm Wheeler-Nicholson wrote an article about all the elements of "Yankee ingenuity" that would allow America to emerge victorious from World War II.
Nicholson correctly identified two of the biggest problems paratroopers face in an assault. First, troops are vulnerable during their descent from hundreds of feet. Second, the soldiers are spread out over a large area by the nature of the drop.
The former cavalry officer suggested that "Bat-Men," paratroopers fitted with special wingsuits that had become popular at airshows, could safely open their parachutes at lower altitudes, making it harder for ground troops to kill the attacking forces. These Bat-Men would also be able to steer themselves in the air, allowing them to land closer together and form up for their assault more quickly.
While Big Army doesn't seem to have ever embraced the idea, the California State Guard created a test unit to try out the suits in 1942. They were led by Mickey Morgan, an air show performer who was famous in the 1940s.
But if the unit ever saw any action, it seems to have been lost to history. While California suffered a few Japanese attacks during World War II, mostly firebombings by balloons and submarine-based aircraft, the fires were either put out by local firefighters or the smokejumpers of the 555th Parachute Infantry Battalion. No "Bat-Men" appear to have flown outside of training.
Oddly enough, Nicholson has a tenuous connection to the famous DC Comics character Batman. After Nicholson resigned his Army commission due to a series of high-profile squabbles with military leadership, he started National Allied Publications.
NAP became Detective Comics at the end of 1936 and Detective Comics released the first issue with the Batman character in 1939. But Nicholson had no part in creating the Dark Knight. He had left the company in 1937.