Can a galaxy splits into multiple galaxies? Is there any naturally occurring event powerful enough that can cause a galaxy regardless of shape and size to divide itself into two or more smaller galaxies?
Small galaxies (say of masses $\sim10^7M_\odot$) can be torn apart by stellar feedback, i.e. the energy provided by exploding stars in a starburst. Whether you want to call the debris galaxies is a matter of opinion, I suppose, but since these guys typically are satellites of larger galaxies, it will probably be torn completely apart by tidal forces, eventually accreting onto the larger one.

Large galaxies (say of masses $\sim10^{11}M_\odot$) are too massive to be split by feedback, but may lose matter (mostly gas, rarely stars) through large-scale outflows created by stellar feedback (or a quasar jet). However, when two large galaxies collide and merge, sometimes large chunk of gas, stars, and dark matter are slung out, which arguably could be called individual galaxies. Most of the matter of the two original galaxies become one, though.