A new realization of extended quantum-mechanical supersymmetry (QM SUSY) with central extension is investigated. We first show that two different sets of d+2 (d+1) supercharges for d=even (odd), each of which satisfies an N=d+2 (d+1) extended QM SUSY algebra without central extension, are hidden in the four-dimensional mass spectrum of the (4+d)-dimensional Dirac action. We then find that the whole set of the supercharges forms an N=2d+4 (2d+2) extended QM SUSY algebra with central charges for d=even (odd). The representation of the supersymmetry algebra is shown to be 1/2-Bogomol'nyi-Prasad-Sommerfield states that correspond to a short representation for the supersymmetry algebra with a central extension. We explicitly examine the four-dimensional mass spectrum of the models with the hyperrectangle and the torus extra dimensions, and discuss their supersymmetric structures. © 2019 authors. Published by the American Physical Society.