  A Hamiltonian two-field gyrofluid model for kinetic Alfv\'en waves (KAWs) in
a magnetized electron-proton plasma, retaining ion finite-Larmor-radius
corrections and parallel magnetic field fluctuations, is used to study the
inverse cascades that develop when turbulence is randomly driven at sub-ion
scales. In the directions perpendicular to the ambient field, the dynamics of
the cascade turns out to be nonlocal and the ratio $\chi_f$ of the wave period
to the characteristic nonlinear time at the driving scale affect some of its
properties. For example, at small values of $\chi_f$, parametric decay
instability of the modes driven by the forcing can develop, enhancing for a
while inverse transfers. The balanced state, obtained at early time when the
two counter-propagating waves are equally driven, also becomes unstable at
small $\chi_f$, leading to an inverse cascade. For $\beta_e$ smaller than a few
units, the cascade slows down when reaching the low-dispersion spectral range.
For higher $\beta_e$, the ratio of the KAW to the Alfv\'en frequencies displays
a local minimum. At the corresponding transverse wavenumber, a condensate is
formed, and the cascade towards larger scales is then inhibited. Depending on
the parameters, a parallel inverse cascade can develop, enhancing the
elongation of the ion-scale magnetic vortices that generically form.
