Context: Unevolved metal-poor stars constitute a fossil record of the early Galaxy, and can provide invaluable information on the properties of the first generations of stars. Binary systems also provide direct information on the stellar masses of their member stars. Aims: The purpose of this investigation is a detailed abundance study of the double-lined spectroscopic binary CS 22876-032, which comprises the two most metal-poor dwarfs known. Methods: We used high-resolution, high-S/N ratio spectra from the UVES spectrograph at the ESO VLT telescope. Long-term radial-velocity measurements and broad-band photometry allowed us to determine improved orbital elements and stellar parameters for both components. We used OSMARCS 1D models and the turbospectrum spectral synthesis code to determine the abundances of Li, O, Na, Mg, Al, Si, Ca, Sc, Ti, Cr, Mn, Fe, Co and Ni. We also used the CO^5BOLD model atmosphere code to compute the 3D abundance corrections, notably for Li and O. Results: We find a metallicity of [Fe/H] ~ -3.6 for both stars, using 1D models with 3D corrections of ~-0.1 dex from averaged 3D models. We determine the oxygen abundance from the near-UV OH bands; the 3D corrections are large, -1 and -1.5 dex for the secondary and primary respectively, and yield [O/Fe] ~ 0.8, close to the high-quality results obtained from the [OI] 630 nm line in metal-poor giants. Other [ α/Fe] ratios are consistent with those measured in other dwarfs and giants with similar [Fe/H], although Ca and Si are somewhat low ([X/Fe] ⪉ 0). Other element ratios follow those of other halo stars. The Li abundance of the primary star is consistent with the Spite plateau, but the secondary shows a lower abundance; 3D corrections are small. Conclusions: The Li abundance in the primary star supports the extension of the Spite Plateau value at the lowest metallicities, without any decrease. The low abundance in the secondary star could be explained by endogenic Li depletion, due to its cooler temperature. If this is not the case, another, yet unknown mechanism may be causing increased scatter in A(Li) at the lowest metallicities.