  We introduce and study a lattice fermion model in one dimension with explicit
$\mathcal{N}=1$ supersymmetry (SUSY). The Hamiltonian of the model is defined
by the square of a supercharge built from Majorana fermion operators. The model
describes interacting Majorana fermions and its properties depend only on a
single parameter $g$. When $g=1$, we find that SUSY is unbroken and the ground
states are identical to those of the frustration-free Kitaev chains. We also
find a parameter regime in which SUSY is restored in the infinite volume limit.
For sufficiently large $g$, we prove that SUSY is spontaneously broken and the
low-lying excitations are gapless, which can be thought of as Nambu-Goldstone
fermions. We then show numerically that these gapless modes have cubic
dispersion at long wavelength.
