Here's a fascinating image of sperm whales "sleeping" vertically in the water, just a few feet below the surface of the sea. It's from a series of photographs provided by National Geographic. (Click the image for a larger view.)
The article cites a 2008 study:
The whales were found to spend seven percent of their day in these vertical sleeping positions near the surface of the water, where they napped from 10 to 15 minutes. Researchers suggested at the time that they might be one of the world's least sleep-dependent animals.
There's more at the link, and more photographs, too, available in a larger size than provided here. It makes very interesting reading.