A new property, analogous to the non-Archimedean property for metrics, is defined for approach convergence spaces (AUCS) and approach uniform spaces. It is proved that the new subcategories (e.g., uAUCS in AUCS) have nice properties: uAUCS is bireflective in AUCS, it is Cartesian closed and contains UCS as a bicoreflective and bireflective subcategory. A comparison to a similar property for approach Cauchy spaces is given: uACHY is bicoreflective in uAUCS.