We observe a density-dependent collective suppression of optical pumping between the hyperfine ground states in an array of submicrometer-sized clouds of cold rubidium atoms. The suppressed Raman transition rate can be explained by strong resonant dipole-dipole interactions that are enhanced by increasing atom density. The observations are consistent with stochastic electrodynamics simulations that incorporate the effects of the nonlinear population transfer via internal atomic levels embedded in a coupled-dipole model.