Animals which are active during the day are termed diurnal.There are many mammals which are diurnal, including: Numbats (the only truly diurnal marsupial) Horses, cattle, sheep and other stock...
Well, there are alot of animals that are nocturnal and that are awake in the day, also alot of animals that switch from one to another. Some examples (that might be true) of animals that stay awake...