Neutron-capture reactions on very neutron-rich nuclei are essential for heavy-element nucleosynthesis through the rapid neutron-capture process, now shown to take place in neutron-star merger events. For these exotic nuclei, radiative neutron capture is extremely sensitive to their γ-emission probability at very low γ energies. In this work, we present measurements of the γ-decay strength of Ni70 over the wide range 1.3≤Eγ≤8 MeV. A significant enhancement is found in the γ-decay strength for transitions with Eγ<3 MeV. At present, this is the most neutron-rich nucleus displaying this feature, proving that this phenomenon is not restricted to stable nuclei. We have performed E1-strength calculations within the quasiparticle time-blocking approximation, which describe our data above Eγ≃5 MeV very well. Moreover, large-scale shell-model calculations indicate an M1 nature of the low-energy γ strength. This turns out to be remarkably robust with respect to the choice of interaction, truncation, and model space, and we predict its presence in the whole isotopic chain, in particular the neutron-rich Ni72,74,76.