The Sun's energy, of course, comes from fusion.
I think there's a small and totally insignificant amount of fission going on as well.
The majority of the Sun's mass is hydrogen, and the vast majority of what isn't hydrogen is helium (with the ratio changing over billions of years as hydrogen is fused into helium). But since the Sun formed from the same nebula as Earth and the other planets, it should contain all the same elements that Earth has. According to this web page, "About 67 elements have been detected in the solar spectrum."
Thus it's very likely that the Sun contains small amounts of uranium-235. The environment is such that it can't form enough of a concentrated mass to trigger a fission chain reaction, but U-235 can, with a low probability, decay by spontaneous fission. Other very heavy isotopes can do the same thing.
I've never heard of any research that indicates that spontaneous fission actually occurs in the Sun, but given its composition it seems almost inevitable that it would, in trivial and insignificant quantities.
For all practical purposes, the answer is no, there is no significant nuclear fission in the Sun. But strictly speaking, yes, there probably is.
Even if it were significant, it wouldn't produce any kind of "perpetual motion". To do that, you'd need, for example, element A fusing into element B, and then element B fissioning back into element A. That can happen (rarely), but it can't produce net energy; if one of the reactions produces energy, the other must absorb energy.
Oh yes, also hydrogen bombs (link is to Martin Beckett's answer). The fission and fusion are both exothermic (i.e., they produce energy), but the fission applies to very heavy elements and the fusion to very light ones.