Irradiation of a sharp tungsten tip by a femtosecond laser and exposed to a strong DC electric field led to gradual and reproducible surface modifications. By a combination of field emission microscopy and scanning electron microscopy, we observed asymmetric surface faceting with sub-ten nanometer high steps. The presence of well pronounced faceted features mainly on the laser-exposed side implies that the surface modification was driven by a laser-induced transient temperature rise -- on a scale of a couple of picoseconds -- in the tungsten tip apex. Moreover, we identified the formation of a nano-tip a few nanometers high located at one of the corners of a faceted plateau. The results of simulations emulating the experimental conditions, are consistent with the experimental observations. The presented conditions can be used as a new method to fabricate nano-tips of few nm height, which can be used in coherent electron pulses generation. Besides the direct practical application, the results also provide insight into the microscopic mechanisms of light-matter interaction. The apparent growth mechanism of the features may also help to explain the origin of enhanced electron field emission, which leads to vacuum arcs, in high electric-field devices such as radio-frequency particle accelerators.