Context: Extremely metal-poor (EMP) stars in the halo of the Galaxy are sensitive probes of the production of the first heavy elements and the efficiency of mixing in the early interstellar medium. The heaviest measurable elements in such stars are our main guides to understanding the nature and astrophysical site(s) of early neutron-capture nucleosynthesis. Aims: Our aim is to measure accurate, homogeneous neutron-capture element abundances for the sample of 32 EMP giant stars studied earlier in this series, including 22 stars with [Fe/H]< -3.0. Methods: Based on high-resolution, high S/N spectra from the ESO VLT/UVES, 1D, LTE model atmospheres, and synthetic spectrum fits, we determine abundances or upper limits for the 16 elements Sr, Y, Zr, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Dy, Ho, Er, Tm, and Yb in all stars. Results: As found earlier, [Sr/Fe], [Y/Fe], [Zr/Fe] and [Ba/Fe] are below Solar in the EMP stars, with very large scatter. However, we find a tight anti-correlation of [Sr/Ba], [Y/Ba], and [Zr/Ba] with [Ba/H] for -4.5 <[Ba/H] < -2.5, also when subtracting the contribution of the main r-process as measured by [Ba/H]. Spectra of even higher S/N ratio are needed to confirm and extend these results below [Fe/H] ? -3.5. The huge, well-characterised scatter of the [n-capture/Fe] ratios in our EMP stars is in stark contrast to the negligible dispersion in the [ ?/Fe] and [Fe-peak/Fe] ratios for the same stars found in Paper V. Conclusions: These results demonstrate that a second (?weak? or LEPP) r-process dominates the production of the lighter neutron-capture elements for [Ba/H] < -2.5. The combination of very consistent [ ?/Fe] and erratic [n-capture/Fe] ratios indicates that inhomogeneous models for the early evolution of the halo are needed. Our accurate data provide strong constraints on future models of the production and mixing of the heavy elements in the early Galaxy. Based on observations made with the ESO Very Large Telescope at Paranal Observatory, Chile (program ID 165.N-0276(A); P.I: R. Cayrel).