  We present high-resolution and high-quality UVES spectroscopic data of the
metal-poor double-lined spectroscopic binary CS 22876--032 ([Fe/H] $\sim -3.7$
dex), with the goal to derive the $^6$Li/$^7$Li isotopic ratio by analysing the
\ion{Li}{i} $\lambda$~670.8~nm doublet. We coadd all 28 useful spectra
normalised and corrected for radial velocity to the rest frame of the primary
star. We fit the Li profile with a grid of the 3D-NLTE synthetic spectra, to
take into account the line profile asymmetries induced by stellar convection,
and perform Monte Carlo simulations to evaluate the uncertainty of the fit of
the Li line profile. We check 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 0.15~dex. The best fit of the Li profile of the primary star
provides A(Li)~$ = 2.17 \pm 0.01$~dex and $^6$Li/$^7$Li~$=8^{+2}_{-5}$\% at
68\% confidence level. In addition, we improve the Li abundance of the
secondary star at A(Li)~$= 1.55 \pm 0.04$~dex, 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 0.5~dex 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 support the need for non standard $^6$Li
production in the early Universe.
