We analyze the polaron in a Wigner lattice, i.e., the interaction of an external electron with electrons in a quasi-two-dimensional Wigner crystal, configured on a dielectric layer with a metallic substrate. Particular attention is paid to the dynamics of the system and to the electron-phonon interaction. The polaron wave function and ground-state energy of the system are calculated in the extended small-polaron theory. The theory is based on the complete set of Wannier functions, which enables us to treat also the polaron dispersion and the first correction to the standard polaron self-energy. We also discuss the T=0 Wigner phase transition, i.e., melting of the electron lattice due to increased electron density. The general agreement with the results obtained previously within the Schrödinger-Rayleigh perturbation theory is good, but also we found some significant differences. The new calculations show that (i) the polaron dispersion is significant at all electron densities and in most cases it resembles the dispersion of lattice electrons; (ii) the critical density parameter rc for a Wigner phase transition in a high density region is close to the value rc≈40 predicted for a strictly two-dimensional Wigner lattice, regardless of the dielectric layer thickness.