We have in detail characterized the anisotropic charge response of the dimer Mott insulator $kappa$-(BEDT-TTF)$_2$-Cu$_2$(CN)$_3$ by dc conductivity, Hall effect and dielectric spectroscopy. At room temperature the Hall coefficient is positive and close to the value expected from stoichiometry; the temperature behavior follows the dc resistivity $rho(T)$. Within the planes the dc conductivity is well described by variable-range hopping in two dimensions; this model, however, fails for the out-of-plane direction. An unusually broad in-plane dielectric relaxation is detected below about 60 K; it slows down much faster than the dc conductivity following an Arrhenius law. At around 17 K we can identify a pronounced dielectric anomaly concomitantly with anomalous features in the mean relaxation time and spectral broadening. The out-of-plane relaxation, on the other hand, shows a much weaker dielectric anomaly; it closely follows the temperature behavior of the respective dc resistivity. At lower temperatures, the dielectric constant becomes smaller both within and perpendicular to the planes; also the relaxation levels off. The observed behavior bears features of relaxor-like ferroelectricity. Because heterogeneities impede its long-range development, only a weak tunneling-like dynamics persists at low temperatures. We suggest that the random potential and domain structure gradually emerge due to the coupling to the anion network.
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