We present the results of a dedicated search for extremely metal-poor stars in the Fornax, Sculptor, and Sextans dSphs. Five stars were selected from two earlier VLT/Giraffe and HET/HRS surveys and subsequently followed up at high spectroscopic resolution with VLT/UVES. All of them turned out to have [Fe/H] ⪉ -3 and three stars are below [Fe/H] ~ -3.5. This constitutes the first evidence that the classical dSphs Fornax and Sextans join Sculptor in containing extremely metal-poor stars and suggests that all of the classical dSphs contain extremely metal-poor stars. One giant in Sculptor at [Fe/H] = -3.96 ± 0.06 is the most metal-poor star ever observed in an external galaxy. We carried out a detailed analysis of the chemical abundances of the α, iron peak, and the heavy elements, and we performed a comparison with the Milky Way halo and the ultra faint dwarf stellar populations. Carbon, barium, and strontium show distinct features characterized by the early stages of galaxy formation and can constrain the origin of their nucleosynthesis. Tables 6 and 7 are only available in electronic form at http://www.aanda.org