Does nuclear fission also happen inside the Sun?

Fission is the splitting of atoms, either by radioactive decay or by collisional impact. Certainly radioactive decay occurs because the sun contains many radioactive isotopes inclusing thorium, uranium etc. Fission of the kind we see in nuclear reactors and bombs is probably rare because the atoms that can actually fuse together are few in number because the density of the sun is so low. It would tale a long time for candidate 'fusable' atoms to come together with exactly the right impact geometry and energy to actually fuse into a new atom. It is not a significant source of energy.