  We construct a mass dimension one fermionic field associated with flag-dipole
spinors. These spinors are related to Elko (flag-pole spinors) by a
one-parameter matrix transformation $\mathcal{Z}(z)$ where $z$ is a complex
number. The theory is non-local and non-covariant. While it is possible to
obtain a Lorentz-invariant theory via $\tau$-deformation, we choose to study
the effects of non-locality and non-covariance. Our motivation for doing so is
explained. We show that a fermionic field with $|z|\neq1$ and $|z|=1$ are
physically equivalent. But for fermionic fields with more than one value of
$z$, their interactions are $z$-dependent thus introducing an additional
fermionic degeneracy that is absent in the Lorentz-invariant theory. We study
the fermionic self-interaction and the local $U(1)$ interaction. In the
process, we obtained non-local contributions for fermionic self-interaction
that have previously been neglected. For the local $U(1)$ theory, the
interactions contain time derivatives that renders the interacting density
non-commutative at space-like separation. We show that this problem can be
resolved by working in the temporal gauge. This issue is also discussed in the
context of gravity.
