Splitting the wavefunction into disjoint pieces sounds conceivable, but that isn't splitting the electron.

The wavefunction describes the probability of finding the electron at each point (there is also a phase factor in there as well), but if you do an experiment to find the electron, it will be exactly at one point. Which point you get depends on the probabilities.

So, if the wavefunction splits into two, e.g., then there is a 50-50 chance of finding it in each piece when you look, but no chance whatsoever of finding half of it in one piece and the other half in the other.

Similarly, when you have an electron in the 1s state, you DON'T have zillions of partial electrons distributed around the region where the wavefunction goes, you have ONE electron which might be at any point in that region.