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. Based on high-resolution, high S/N spectra from the ESO VLT/U
VES, 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. 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]. 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 [$alpha$/Fe] and [Fe-peak/Fe] ratios for the same stars found in Paper V. 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 [$alpha$/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.
