  We discuss the dissipative preparation of p-wave superconductors in
number-conserving one-dimensional fermionic systems. We focus on two setups:
the first one entails a single wire coupled to a bath, whereas in the second
one the environment is connected to a two-leg ladder. Both settings lead to
stationary states which feature the bulk properties of a p-wave superconductor,
identified in this number-conserving setting through the long-distance behavior
of the proper p-wave correlations. The two schemes differ in the fact that the
steady state of the single wire is not characterized by topological order,
whereas the two-leg ladder hosts Majorana zero modes, which are decoupled from
damping and exponentially localized at the edges. Our analytical results are
complemented by an extensive numerical study of the steady-state properties, of
the asymptotic decay rate and of the robustness of the protocols.
