A new study shows that frigatebirds can--and do--sleep while flying based on their brain signals, though the amount of sleep in the air is much less than on land.

Some birds are known to fly continuously for long periods, but scientists had not been able to prove their ability to sleep while flying until now. 
An international team of researchers made the determination by equipping frigatebirds on Galápagos Island with small devices that measure brain activity and head movements. 
Frigatebirds, according to a news release “...are large seabirds that spend weeks flying non-stop over the ocean in search of flying fish and squid...”
In fact, some of the participant birds were shown to have flown 1,800-plus miles for up to ten days. 
Based on brain activity, the team found that the frigatebirds "...sleep in flight with either one cerebral hemisphere at a time or both hemispheres simultaneously. Despite being able to engage in all types of sleep in flight, the birds slept less than an hour a day, a mere fraction of the time spent sleeping on land."
Scientists are still not sure how they are able to function “on such little sleep.” 
This is one of the areas that will likely be explored by the researchers in the future.