The effect of nanoscale confinement on the local polymer dynamics has been studied by means of dielectric spectroscopy in the frequency range from 10 mHz to 1 MHz. We have used microphase-separated PS/LC block copolymers (PS = polystyrene, LC = liquid crystalline). Depending on volume fraction, the LC blocks were confined to a layer with thickness of 21.3 and 11.2 nm lamellae morphology) or were contained in domains of cylindrical or spherical form with diameter of 12.6 and 9.4 nm, respectively. At lower concentration of PS blocks, the LC copolymer form a continuous LC matrix with cylindrical inclusions of PS component. For all morphologies the LC block reveals two dielectric relaxations related to the local motion of mesogen (beta process) and spacer (gamma process). The relaxation time as well as the activation energy for the gamma process was independent of spatial constraints even for the smallest characteristic length of 9.4 nm. The P relaxation, however, speeds up, and its activation energy decreases for confinement lengths less than 20 nm. The gamma process has an essentially noncooperative nature, whereas the beta process exhibits a positive apparent activation entropy, reflecting a partial cooperative feature. As a result, the beta relaxation is affected by confinement just as the essentially cooperative dynamic glass transition (a relaxation) in polymers.