Fast charging of Li-ion batteries would ease consumer concerns regarding refueling time of electric vehicles. In this series of articles, we are systematically examining the effects of high currents flowing through the cell. Here we consider open-circuit relaxation dynamics for cell voltage and individual electrode potentials after abrupt termination of steady-state charging currents (1-6C) or short current pulses (< 30 s) of either polarity (to 16C). We demonstrate that dispersive kinetics observed after these current perturbations become faster for higher currents applied before the rest period. Our observations indicate that the voltage relaxation behaviors (i) depend on the current during charge, and less so on the state of charge, (ii) display universal stretched-exponential kinetics, (iii) can be induced by a short current pulse, (iv) require a minimum amount of charge transfer during the pulse, and (v) originate mainly from the oxide cathode. Trap controlled Li+ ion transport through a partially disordered oxygen-depleted subsurface layer in the cathode is suggested as a possible origin for the relaxation behavior. The slow multidecadal responses to high currents can cause the nonlinearities observed during rapid charging of the electrochemical cells, reported in Part 2 of this series. (C) The Author(s) 2019. Published by ECS.