A novel enhancement of drop and spray cooling for microelectronic and radiological elements and server rooms requiring extremely high heat fluxes is proposed. The key idea of the method is to cover the heat transfer surfaces with electrospun non-woven polymer nanofiber mats. The mats are permeable for water drops. The enhanced efficiency of drop cooling in the presence of nanofiber mats observed experimentally results from full elimination of receding and bouncing of the drops, characteristic of the current spray cooling technology. Therefore, the drops evaporate completely, and the large cooling potential associated with the latent heat of water evaporation is more fully exploited. This is paradoxical: the best cooling can be provided by a “fur overcoat”! The proposed cooling method alone may lead to a breakthrough in further miniaturization of microelectronic chips, optical and radiological elements and accelerate the development of a new generation of computers. In order to check the suitability of different materials for the drop and spray cooling applications, the thermal and structural properties of nanofiber mats based on four different polymers have been measured over a wide temperature range. Based on the results of these measurements, the most suitable materials have been chosen.