A variety of two-dimensional materials possess a band structure with an energy extremal ridge along a ring in momentum space. Examples are biased bilayer graphene, and surfaces and interfaces with a Rashba spin-orbit interaction where at low doping the carriers fill an annulus... This topological feature causes an anomalous screening behavior, which we study using the Thomas-Fermi theory. Specifically, reducing the doping is predicted to enhance the linear screening response, while at zero doping the size of the screening cloud surrounding a Coulomb impurity is found to increase as the cube root of the impurity charge. (read more)