  We introduce a non-linear extension of Proca's field theory for massive
vector (spin $1$) bosons. The associated relativistic nonlinear wave equation
is related to recently advanced nonlinear extensions of the Schroedinger,
Dirac, and Klein-Gordon equations inspired on the non-extensive generalized
thermostatistics. This is a theoretical framework that has been applied in
recent years to several problems in nuclear and particle physics, gravitational
physics, and quantum field theory. The nonlinear Proca equation investigated
here has a power-law nonlinearity characterized by a real parameter $q$
(formally corresponding to the Tsallis entropic parameter) in such a way that
the standard linear Proca wave equation is recovered in the limit $q
\rightarrow 1$. We derive the nonlinear Proca equation from a Lagrangian that,
besides the usual vectorial field $\Psi^{\mu}(\vec{x},t)$, involves an
additional field $\Phi^{\mu}(\vec{x},t)$. We obtain exact time dependent
soliton-like solutions for these fields having the form of a $q$-plane wave,
and show that both field equations lead to the relativistic energy-momentum
relation $E^{2} = p^{2}c^{2} + m^{2}c^{4}$ for all values of $q$. This suggests
that the present nonlinear theory constitutes a new field theoretical
representation of particle dynamics. In the limit of massless particles the
present $q$-generalized Proca theory reduces to Maxwell electromagnetism, and
the $q$-plane waves yield localized, transverse solutions of Maxwell equations.
Physical consequences and possible applications are discussed.
