We report on a survey of $^7$Li and $^6$Li isotopic abundances in metal-poor halo stars. The spectra of the 24 stars observed with VLT/UVES are of exceptionally high quality: $S/N>400$ and resolving power $R \simeq 120 000$. The $^7$Li abundances on our H$\alpha$$T_{\rm eff}$-scale show very small intrinsic scatter and a pronounced [Fe/H]-dependence. The resulting estimated primordial $^7$Li abundance is about 0.5 dex lower than predicted from Big Bang nucleosynthesis and the baryon density inferred by the cosmic microwave background. Nine of the stars yield a positive detection (${>}2\sigma$) of $^6$Li, which suggests the existence of a $^6$Li plateau for halo stars. The most interesting result is the presence of $^6$Li in the very metal-poor ([Fe/H]$=-2.74$) dwarf LP815-43 at the level of $^6$Li/$^7$Li${\simeq}\, 0.05\pm0.02$. According to models for stellar Li depletion due to diffusion or rotationally-induced mixing, a 0.5 dex $^7$Li depletion would require an unrealistic high initial $^6$Li abundance ($\log ^6{\rm Li} \ge 2.0$). Simultaneously, the observed high $^6$Li abundance at such low [Fe/H] can not be reconciled with existing models for Galactic cosmic ray spallation and $\alpha$-fusion reactions. This opens up exciting prospects of pre-Galactic $^6$Li production, possibly due to cosmological cosmic rays or late-decaying massive particles such as the gravitino or neutralino in the Big Bang.