The resistivity and the Hall effect as a function of current density and temperature in the normal state of very thin films of YBa 2 Cu 3 O 6+x are measured with a fast pulsedcurrent technique. The in-plane longitudinal and transverse (Hall) conductivities are reduced in intense current densities up to several MA cm −2 . In optimally-doped samples, this non-ohmic effect is limited to temperatures below 100 K. In a moderately underdoped sample, however, the non-linearity extends to the temperature range from T c = 53 K to ∼150 K. The non-linear part of the Hall conductivity does not scale with a critical exponent, thus excluding classical amplitude fluctuations of the superconducting order parameter as possible origin.