We predict nanoscale field and dipole patterns due to the broken uniformity of a laser-driven local field in 1D and 2D lattices. They may result in size-related resonances and large field enhancement, which in turn can give rise to low-intensity nonlinear optical effects, e.g., optical bistability, even in the ultimate case of a pair of coupled atoms. At certain, "magic" numbers and configurations of atoms in a lattice, the system may exhibit the self-induced cancellation of the suppression of a local field.