No, Cleo!

Unicorns are equine animals with a horn on their forehead. This horn gives them a + 15 dressage bonus.

They are bred in a distinctive way:

• both parents must be unicorns of the same breed
• 1 covering session must be available

then you'll have a 2 in 5 chance of getting a unicorn foal.

After the same unicorn mare foals 5 times without giving birth to a unicorn foal, the 6ᵗʰ is guaranteed to be a unicorn foal.

After covering a unicorn mare, you must wait 10 hours in real-time before you can cover her again or before you can cover a potential unicorn filly.
Draft unicorns can breed up to 3 times during their lifetime.