Be like the Majestic Dolphin.
The solution is to have the animal evolve to sleep like a dolphin, with half their brain at a time, though for different reasons. Dolphins on Earth do this to avoid drowning, but the concept is still the same. Without a method like this, your people will have a hard time living without sleep related issue, like the dolphins said "So long and thanks for all the fish, so sad that it must come to this."
This opens the door to some interesting psychological and physiological features;
- For one, growth hormones are released during sleep, so this species may grow at half the rate of humans but for 24 hours a day.
- I imagine as individual sections of brain controlling movement rest, certain body parts may be paralyzed at certain parts of the day.
- Depending on the sections of brain, they could possibly have different personalities, depending on what half was awake.
When you get started here, you can really get rolling. I can imagine a species that instead of sleeping half a brain at a time; lives on quarters, or even eighths at a time. Keep in mind that if they evolved in this world, they wouldn't even have a concept of night; never mind associate it with sleep. Of course, if you have your world suddenly have no night, then this answer is pointless.