  In this work we discuss the properties of a modified Born-Infeld
electrodynamics in the framework of very special relativity (VSR). This
proposal allows us to study VSR mass effects in a gauge-invariant context of
nonlinear electrodynamics. It is analyzed in detail the electrostatic solutions
for two different cases, as well as the VSR dispersion relations are found to
be of a \emph{massive} particle with nonlinear modifications. Afterwards, the
field energy and static potential are computed, in the latter we find from the
VSR contribution a novel long-range $1/L^3$ correction to the Coulomb
potential, in contrast to the $1/L^5$ correction of the usual Born-Infeld
theory.
