While large scale primordial non-Gaussianity is strongly constrained by present-day data, there are no such constraints at Mpc scales. Here we investigate the effect of significant small-scale primordial non-Gaussianity on structure formation and the galaxy formation process with collisionless simulations: specifically, we explore four different types of non-Gaussianities. Generically, we find a distinct and potentially detectable feature in the matter power spectrum around the non-linear scale. We then show in particular that a negatively-skewed distribution of the potential random field, hence positively skewed in terms of overdensities, with $f_{\rm NL} \approx -1000$ at these scales, implies that typical galaxy-sized halos reach half of their present-day mass at an earlier stage and have a quieter merging history than in the Gaussian case. Their environment between 1 and 5 virial radii at $z=0$ is less dense than in the Gaussian case. This quieter history and less dense environment has potentially interesting consequences in terms of the formation of bulges and bars. Moreover, we show that subhalos have a more flattened distribution around their host than in the Gaussian case, albeit not as flattened as the 11 most massive Milky Way satellites, and that the two most massive subhalos tend to display an interesting anti-correlation of velocities around their host, indicative of kinematic coherence. All these hints will need to be statistically confirmed in larger-box simulations with scale-dependent non-Gaussian initial conditions, followed by hydrodynamical zoom-in simulations to explore the detailed consequences of small-scale non-Gaussianities on galaxy formation.