  Charged Lifshitz black holes for the Einstein-Proca-Maxwell system with a
negative cosmological constant in arbitrary dimension $D$ are known only if the
dynamical critical exponent is fixed as $z=2(D-2)$. In the present work, we
show that these configurations can be extended to much more general charged
black holes which in addition exist for any value of the dynamical exponent
$z>1$ by considering a nonlinear electrodynamics instead of the Maxwell theory.
More precisely, we introduce a two-parametric nonlinear electrodynamics defined
in the more general, but less known, so-called $(\mathcal{H},P)$-formalism and
obtain a family of charged black hole solutions depending on two parameters. We
also remark that the value of the dynamical exponent $z=D-2$ turns out to be
critical in the sense that it yields asymptotically Lifshitz black holes with
logarithmic decay supported by a particular logarithmic electrodynamics. All
these configurations include extremal Lifshitz black holes. Charged topological
Lifshitz black holes are also shown to emerge by slightly generalizing the
proposed electrodynamics.
