  Lattice models with supersymmetry are known to exhibit a variety of
remarkable properties that do not exist in the relativistic models. In this
paper, we introduce an interacting generalization of the Kitaev chain of
Majorana fermions with $\mathcal{N} = 1$ supersymmetry and investigate its
low-energy properties, paying particular attention to the ground-state
degeneracy and low-lying fermionic excitations. First, we establish the
existence of a phase with spontaneously broken supersymmetry and a phase
transition out of it with the help of variational arguments and the exact
ground state. We then develop, based on the superfield formalism, a simple
mean-field theory, in which the order parameters detect supersymmetry-breaking,
to understand the ground-state phases and low-lying Nambu-Goldstone fermions.
At the solvable point ({\em frustration-free point}), the exact ground state of
an open chain exhibits large degeneracy of the order of the system size, which
is attributed to the existence of a zero-energy domain wall (dubbed kink or
skink) separating the topological and trivial states of Majorana fermions. Our
results may shed new light on the intriguing ground-state properties of
supersymmetric lattice models.
