Do we know of a celestial body which doesn't spin at all on its own axis? If not, why is it so?
Many red and brown dwarfs must have planets that are in full tidal lock with their host star, that means that the same side of the planet is always facing the star so there is always day while on the other side there is always night. Our Moon and many other moons are also in such a tidal lock with their planet but not with the Sun so that the often-used term "Dark side of the Moon" is verbally incorrect since on the far side there is a day too (when on Earth it is a New Moon).
If you mean non-spinning in the sense that on the same side of the planet it is day for half a year and night the other half I'm quite sure such planets exist too. On SpaceEngine I've seen similar planets. Nothing speaks against the existence of such planets I believe. There also might exist moons whose same side faces their planet for half a year and the other half faces away from it.
Every object in the universe--from a basic star to an exotic black hole--spins, and the origin of that spin can be traced back to the very beginning of time. Within instants after the Big Bang, the primordial fireball of energy expanded at an incredible speed, then later cooled and solidified into all the matter in the universe. Had this fireball been uniform in all directions, everything we see today would be completely homogeneous.