Cosmic background neutrinos have a large velocity dispersion, which causes the evolution of long-wavelength density perturbations to depend on scale. This scale-dependent growth leads to the well-known suppression in the linear theory matter power spectrum that is used to probe neutrino mass. In this paper, we study the impact of long-wavelength density perturbations on small-scale structure formation. By performing separate universe simulations where the long-wavelength mode is absorbed into the local expansion, we measure the responses of the cold dark matter (CDM) power spectrum and halo mass function, which correspond to the squeezed-limit bispectrum and halo bias. We find that the scale-dependent evolution of the long-wavelength modes causes these quantities to depend on scale and provide simple expressions to model them in terms of scale and the amount of massive neutrinos. Importantly, this scale-dependent bias reduces the suppression in the linear halo power spectrum due to massive neutrinos by 13% and 26% for objects of bias b ¯=2 and b ¯≫1 , respectively. We demonstrate with high statistical significance that the scale-dependent halo bias cannot be modeled by the CDM and neutrino density transfer functions at the time when the halos are identified. This reinforces the importance of the temporal nonlocality of structure formation, especially when the growth is scale dependent.