We calculate the hydrodynamic forces exerted on an oscillating circular cylinder when it moves translationally perpendicular to its axis in the infinitely deep water covered by compressed ice. The cylinder can oscillate both horizontally and vertically. In the linear approximation, we find a solution for the steady wave motion generated by the cylinder within the hydrodynamic set of equations for the incompressible ideal fluid. We show that depending on the rate of ice compression, the normal and anomalous dispersion can occur in the system. In the latter case, the group velocity can be opposite to the phase velocity in a certain range of wavenumbers. We investigate the dependences of the hydrodynamic loads (added mass, damping coefficients, wave resistance, and lift force) exerting on the cylinder on the translational velocity and frequency of oscillation. It is shown that there is a possibility of the appearance of negative values for the damping coefficients at the relatively big cylinder velocity; then the wave resistance decreases with increasing of cylinder velocity. The theoretical results are underpinned by the numerical calculations for the real parameters of ice and cylinder motion.