Using porous electrodes containing redox-active nickel hexacyanoferrate (NiHCF) nanoparticles, we construct and test a device for capacitive deionization in a two flow-channel device where the intercalation electrodes are in direct contact with an anion-exchange membrane. Upon reduction of NiHCF, cations intercalate into it and the water in its vicinity is desalinated; at the same time water in the opposing electrode becomes more saline upon oxidation of NiHCF in that electrode. In a cyclic process of charge and discharge, fresh water is continuously produced, alternating between the two channels in sync with the direction of applied current. We present proof-of-principle experiments of this technology for single salt solutions, where we analyze various levels of current and cycle durations. We analyze salt removal rate and energy consumption. In desalination experiments with salt mixtures we find a threefold enhancement for K+ over Na+-adsorption, which shows the potential of NiHCF intercalation electrodes for selective ion separation from mixed ionic solutions.