  The future Electron - Ion ($eA$) Collider is expected to probe the high
energy regime of the QCD dynamics, with the exclusive vector meson production
cross section being one of the most promising observables. In this paper we
complement previous studies of exclusive processes presenting a comprehensive
analysis of diffractive $\rho$ production at small $x$. We compute the coherent
and incoherent cross sections taking into account non-linear QCD dynamical
effects and considering different models for the dipole - proton scattering
amplitude and for the vector meson wave function. The dependence of these cross
sections with the energy, photon virtuality, nuclear mass number and squared
momentum transfer is analysed in detail. Moreover, we compare the non-linear
predictions with those obtained in the linear regime. Finally, we also estimate
the exclusive photon, $J/\Psi$ and $\phi$ production and compare with the
results obtained for $\rho$ production. Our results demonstrate that the
analysis of diffractive $\rho$ production in future electron - ion colliders
will be important to understand the non-linear QCD dynamics.
