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Stationary and nonstationary spatiotemporal pattern formations emerging from the cellular electric activity are a common feature of biological cells and tissues. The nonstationary ones are well explained in the framework of the cable model. Inversely, the formation of the widespread self-organized stationary patterns of transcellular ionic currents remains elusive, despite their importance in cell polarization, apical growth, and morphogenesis. For example, the nature of the breaking symmetry in the Fucus zygote, a model organism for the experimental investigation of embryonic pattern formation, is still an open question. Using an electrodiffusive model, we report here an unexpected property of the cellular electric activity: a phase-space domain that gives rise to stationary patterns of transcellular ionic currents at finite wavelength. The cable model cannot predict this instability. In agreement with experiments, the characteristic time is an ionic diffusive one (<2 min). The critical radius is of the same order of magnitude as the cell radius (30 microm). The generic salient features are a global positive differential conductance, a negative differential conductance for one ion, and a difference between the diffusive coefficients. Although different, this mechanism is reminiscent of Turing instability.