The authors extend to higher dimensions the known two-dimensional result of S. E. Newhouse [Topology 13, 9-18 (1974; Zbl 0275.58016)] in the following main Theorem: Near any smooth diffeomorphism exhibiting a homoclinic tangency associated to a sectionally dissipative saddle, there is a residual subset of an open set of diffeomorphisms such that each of its elements displays infinitely many coexisting sinks (attracting periodic orbits).