We present high-resolution and high-quality UVES spectroscopic data of the metal-poor double-lined spectroscopic binary CS 22876-032 ([Fe/H] approximately -3.7dex). Our goal is to derive the ^6^Li/^7^Li isotopic ratio by analysing the LiI 670.8nm doublet. Methods. We co-added all 28 useful spectra normalised and corrected for radial velocity to the rest frame of the primary star. We fitted the Li profile with a grid of the 3D non-local thermodynamic equilibrium (NLTE) synthetic spectra to take into account the line profile asymmetries induced by stellar convection, and performed Monte Carlo simulations to evaluate the uncertainty of the fit of the Li line profile. We checked that the veiling factor does not affect the derived isotopic ratio, ^6^Li/^7^Li, and only modifies the Li abundance, A(Li), by about 0.15dex. The best fit of the Li profile of the primary star provides A(Li)=2.17+/-0.01dex and ^6^Li/^7^Li=8^+2^_-5_% at 68% confidence level. In addition, we improved the Li abundance of the secondary star at A(Li)=1.55+/-0.04dex, which is about 0.6 dex lower than that of the primary star. The analysis of the Li profile of the primary star is consistent with no detection of ^6^Li and provides an upper limit to the isotopic ratio of ^6^Li/^7^Li<10% at this very low metallicity, about 0.5dex lower in metallicity than previous attempts for detection of ^6^Li in extremely metal poor stars. These results do not solve or worsen the cosmological ^7^Li problem, nor do they support the need for non-standard ^6^Li production in the early Universe.