  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
$\mathcal{N}=d+2$ ($d+1$) extended QM SUSY algebra without central extension,
are hidden in the four-dimensional (4D) mass spectrum of the
$(4+d)$-dimensional Dirac action. We then find that the whole set of the
supercharges forms an $\mathcal{N}=2d+4$ ($2d+2$) 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 central
extension. We explicitly examine the 4D mass spectrum of the models with the
hyperrectangle and the torus extra dimensions, and discuss their supersymmetric
structures.
