  The damping of the oscillations of a small permanent magnet (spherical shape,
radius 0.1 mm) levitating between two parallel epitaxial YBCO films is measured
as a function of oscillation amplitude and temperature. At small amplitudes the
dissipation is found to be orders of magnitude lower than in bulk YBCO,
Q-factors exceeding one million at low temperatures. With increasing amplitude
the dissipation becomes exponentially large, exceeding the bulk values at large
drives. We describe our results by calculating the ac shielding currents
flowing through trapped flux whose motion gives rise to electric fields. We
find dissipation to originate from different mechanisms of flux dynamics.
