A Faradaic process does not follow Ohm's law. There is an ohmic resistance in the cell of course, due to contacts, concentrations and so on.. but that's only part of the story. The rate of the electrochemical reactions at the surface determines the current and the rate depends exponentially on the potential. You can look up Butler-Volmer to get a description.
However, the reactions can only proceed if there are electroactive species available at the electrode surface. So, if you push the reaction faster than material can be brought to the surface, you will get a mass transport limited current.
Sorry, to go a little of-topic.