A novel nonlinear effect of anomalously deep penetration of an external radio-frequency electric field into a plasma is described. A self-consistent kinetic treatment reveals a transition region between the sheath and the plasma. Because of the electron velocity modulation in the sheath, bunches in the energetic electron density are formed in the transition region adjacent to the sheath. The width of the region is of order V(T)/omega, where V(T) is the electron thermal velocity, and omega is the frequency of the electric field. The presence of the electric field in the transition region results in a collisionless cooling of the energetic electrons and an additional heating of the cold electrons.