No. Electrons cannot split, as far as we can tell. There is an analogous statement to the one which you describe, but for waves. It's called Huygens Principle. To really understand interference, think about the electron's wavefunction as a measure of its probability. It tells you where you may find the electron. Then when you measure you collapse the wavefunction to be in one of its many possibilities, and that's where you see an electron particle.

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