Spacecraft observations in the solar wind and in the Earth’s magnetosheath indicate that the dissipation range of magnetic turbulence probably takes place at electron scales. We derive nonlinear electron magnetohydrodynamic (EMHD) equations for warm plasma, when the ratio of thermodynamic and magnetic pressures is close to 1. This model describes plasma turbulence under the solar wind and magnetosheath conditions on the electron spatial scales and with the characteristic frequency that does not exceed the electron gyrofrequency. At electron scales and in the presence of a sufficiently large temperature anisotropy, when the perpendicular electron temperature exceeds the parallel one, there exist self-organized, coherent, nonlinear dipole vortex structures associated with obliquely propagating whistler waves. These can be visualized as pairs of counterstreaming helicoidal currents that produce both the compressional and torsional perturbations of the magnetic field. This novel solution is an evanescent mode, strongly localized in space that can constitute a building block for the plasma turbulence at short scales and provide a possible scenario of turbulence dissipation at electron scales.