  We present the derivation of a supersymmetric model for fermionic fields with
integer valued mass dimension based on a general superfield with one free
spinor index. First, we demonstrate that it is impossible to formulate such a
model based on a general scalar superfield. This is due to problems
constructing a Lagrangian containing a kinetic term for the fermionic mass
dimension one field, as well as problems deriving a consistent second
quantisation. We then develop a formalism based on a general superfield with
one free spinor index. We systematically derive all associated chiral and
anti-chiral superfields up to third order in covariant derivatives. Using this
formalism we are able to construct a supersymmetric on-shell Lagrangian that
contains a kinetic term for the fermionic fields with mass dimension one. We
then derive the corresponding on-shell supercurrent and succeed to formulate a
consistent second quantisation for the component fields. Finally, we present
our result for a supersymmetric Hamiltonian. As the Lagrangian is by
construction supersymmetric and the Hamiltonian was derived from the Lagrangian
using the supersymmetry algebra the Hamiltonian must be positive definite.
