Context. The primordial nature of the Spite plateau is at odds with the WMAP satellite measurements, implying a primordial Li production at least three times higher than observed. It has also been suggested that A(Li) might exhibit a positive correlation with metallicity below [Fe/H] ~ -2.5. Previous samples studied comprised few stars below [Fe/H] = -3. Aims: We present VLT-UVES Li abundances of 28 halo dwarf stars between [Fe/H] = -2.5 and -3.5, ten of which have [Fe/H] <-3. Methods: We determined stellar parameters and abundances using four different Teff scales. The direct infrared flux method was applied to infrared photometry. Hα wings were fitted with two synthetic grids computed by means of 1D LTE atmosphere models, assuming two different self-broadening theories. A grid of Hα profiles was finally computed by means of 3D hydrodynamical atmosphere models. The Li i doublet at 670.8 nm has been used to measure A(Li) by means of 3D hydrodynamical NLTE spectral syntheses. An analytical fit of A(Li)3D, NLTE as a function of equivalent width, Teff, log g, and [Fe/H] has been derived and is made available. Results: We confirm previous claims that A(Li) does not exhibit a plateau below [Fe/H] = -3. We detect a strong positive correlation with [Fe/H] that is insensitive to the choice of Teff estimator. From a linear fit, we infer a steep slope of about 0.30 dex in A(Li) per dex in [Fe/H], which has a significance of 2-3σ. The slopes derived using the four Teff estimators are consistent to within 1σ. A significant slope is also detected in the A(Li)-Teff plane, driven mainly by the coolest stars in the sample (Teff < 6250), which appear to be Li-poor. However, when we remove these stars the slope detected in the A(Li)-[Fe/H] plane is not altered significantly. When the full sample is considered, the scatter in A(Li) increases by a factor of 2 towards lower metallicities, while the plateau appears very thin above [Fe/H] = -2.8. At this metallicity, the plateau lies at = 2.199±0.086. Conclusions: The meltdown of the Spite plateau below [Fe/H] ~ -3 is established, but its cause is unclear. If the primordial A(Li) were that derived from standard BBN, it appears difficult to envision a single depletion phenomenon producing a thin, metallicity independent plateau above [Fe/H] = -2.8, and a highly scattered, metallicity dependent distribution below. That no star below [Fe/H] = -3 lies above the plateau suggests that they formed at plateau level and experienced subsequent depletion. Based on observations made with the ESO Very Large Telescope at Paranal Observatory, Chile (Programmes 076.A-0463 and 077.D-0299).Full Table 3 is available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/522/A26IDL code (appendix) is only available in electronic form at http://www.aanda.org