We introduce neighborhood models that account for local spatial effects in Galam\\\\\\\'s model of minority opinion spreading. For systems of different sizes N, the time to reach consensus is shown to scale with log(N) in Galam\\\\\\\'s model, while it grows linearly with N in the new neighborhood models. The threshold value of the initial concentration of minority supporters for the defeat of the initial majority, which is independent of N in Galam\\\\\\\'s model, goes to zero with growing system size in the neighborhood models. This is a consequence of the existence of a critical size for the growth of a local domain of minority supporters.