In this paper, we will use implicit functions to obtain a general result about the existence of a unique common fixed point for commuting mappings in ultrametric spaces. This result enables us to improve some known fixed point theorems and enables us to obtain a relation between completeness and the existence of a unique fixed point for self-mappings in non-Archimedean metric spaces. By presenting some counterexamples, we will show that our results cannot be extended to general metric spaces.