Context. The rapid neutron-capture process, which created about half of the heaviest elements in the solar system, is believed to have been unique. Many recent studies have shown that this uniqueness is not true for the formation of lighter elements, in particular those in the atomic number range 38 < Z < 48. Among these, palladium (Pd) and especially silver (Ag) are expected to be key indicators of a possible second r-process, but until recently they have been studied only in a few stars. We therefore target Pd and Ag in a large sample of stars and compare these abundances to those of Sr, Y, Zr, Ba, and Eu produced by the slow (s-) and rapid (r-) neutron-capture processes. Hereby we investigate the nature of the formation process of Ag and Pd. Aims. We study the abundances of seven elements (Sr, Y, Zr, Pd, Ag, Ba, and Eu) to gain insight into the formation process of the elements and explore in depth the nature of the second r-process. Methods. By adopting a homogeneous one-dimensional local thermodynamic equilibrium (1D LTE) analysis of 71 stars, we derive stellar abundances using the spectral synthesis code MOOG, and the MARCS model atmospheres. We calculate abundance ratio trends and compare the derived abundances to site-dependent yield predictions (low-mass O-Ne-Mg core-collapse supernovae and parametrised high-entropy winds), to extract characteristics of the second r-process. Results. The seven elements are tracers of different (neutron-capture) processes, which in turn allows us to constrain the formation process(es) of Pd and Ag. The abundance ratios of the heavy elements are found to be correlated and anti-correlated. These trends lead to clear indications that a second/weak r-process, is responsible for the formation of Pd and Ag. On the basis of the comparison to the model predictions, we find that the conditions under which this process takes place differ from those for the main r-process in needing lower neutron number densities, lower neutron-to-seed ratios, and lower entropies, and/or higher electron abundances. Conclusions. Our analysis confirms that Pd and Ag form via a rapid neutron-capture process that differs from the main r-process, the main and weak s-processes, and charged particle freeze-outs. We find that this process is efficiently working down to the lowest metallicity sampled by our analysis ([Fe/H] = −3.3). Our results may indicate that a combination of these explosive sites is needed to explain the variety in the observationally derived abundance patterns.