Is there any case reported that seems to resemble the following: there is a particle and at some moment, the particle seems to break down into two or more particles that are all identical to the original particle?
In "normal" cases, no, this is not possible. You can easily understand why by considering this process in the center-of-mass frame (which is the rest frame of the original particle). In this frame, you would start with a single particle $X$ at rest, which has energy $m_Xc^2$, and wind up with 2 or more $X$, which will necessarily have an energy of at least $2m_Xc^2$. So energy conservation has to be violated by these sorts of reactions.
But consider a caveat: what do I mean by "normal"? Well, what we consider normal matter is made of massive particles. If you're looking at massless particles, on the other hand, the above argument doesn't apply because a massless particle doesn't have a rest frame. So you have to examine it from a lab frame (that is, any inertial frame). It should be easy to convince yourself that a massless particle of energy $E$ can decay into multiple instances of the same kind of particle with energies $\{E_1,\ldots,E_N\}$ such that $\sum_i E_i = E$ as long as all the products have momentum parallel to that of the original particle.
This type of process is called collinear branching and it actually does have to be taken into account when doing calculations in quantum field theory involving photons and gluons, the two known massless particles. Being "taken into account" is a little different from actually "occurring," though. The effect of this is not to say that photons and gluons actually split into multiples, but that the only meaningful things we can calculate are quantities which don't depend on whether the bosons are splitting or not. So the question of whether or not this decay into copies actually happens cannot be experimentally decided, according to QFT.
