AC losses in the superconducting parts of tapes and multifilamentary coated conductors that are subjected to AC magnetic fields are an important component of the total losses in such composites. The analytical expression for AC losses in a thin superconducting strip with a power-law voltage-current characteristic and critical current depending on a magnetic field is obtained for the case of asymptotically high perpendicular magnetic fields. The losses caused by closure currents are estimated. The results show that the Bean model gives significantly understated values for coated conductors. The applicability of the obtained expressions is analyzed.