We present VLT-UVES Li abundances for 28 halo dwarf stars between [Fe/H]=-2.5 and -3.5, 10 of which have [Fe/H]<-3. Four different T_eff scales have been used. Direct Infrared Flux Method (IRFM) has been used on the basis of 2MASS infrared photometry. H_alpha wings have been fitted against synthetic grids computed by means of 1D LTE atmosphere models, assuming different self-broadening theories. Finally, a grid of H_alpha profiles has been 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, T_eff, log g, and [Fe/H] has been derived and is made available. A(Li) does not exhibit a plateau below [Fe/H]=-3. A strong positive correlation with [Fe/H] appears, not influenced by the choice of the T_eff estimator. From a linear fit, we obtain a strong slope of about 0.30 dex in A(Li) per dex in [Fe/H], significant to 2-3 sigma, and consistent among all the four T_eff estimators. A significant slope is also detected in the A(Li)--T_eff plane, driven mainly by the coolest stars in the sample which appear Li-poor. Removing such stars does not alter the behavior in the A(Li)-[Fe/H] plane. 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. The meltdown of the Spite plateau below [Fe/H]\sim-3 is established, but its cause is unclear. If the primordial A(Li) is the one derived from WMAP, 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. The fact that no star below [Fe/H]=-3 lies above the plateau suggests that they formed at plateau level and underwent subsequent depletion.