Local correlation tracking techniques are used to measure proper motions in a series of high angular resolution (~0.1") penumbra images. If these motions trace true plasma motions, then we have detected converging flows that arrange the plasma in long narrow filaments cospatial with dark penumbral filaments. Assuming that these flows are stationary, the vertical stratification of the atmosphere and the conservation of mass suggest downflows in the filaments on the order of 200 m s-1. The association between downflows and dark features may be a sign of convection, as it happens with the nonmagnetic granulation. Insufficient spatial resolution may explain why the estimated vertical velocities are not fast enough to supply the radiative losses of penumbrae.