Birds that fly long distances sleep with half their brain at a time so they don't fall out of the sky to their doom into the ocean below.

https://www.audubon.org/news/scientists-finally-have-evidenc...

In some old Western movie -- as in The Wild West with cowboys -- there was a Samurai character who claimed to sleep while they marched somewhere.

I have no idea how to look up a citation supporting the idea that this was a real phenomenon, but I did find this article that I kind of skimmed and I don't think it says anything on the topic I want a citation for and additional googling gets me absolutely nothing of value:

https://www.bbc.com/future/article/20160506-the-japanese-art...