The main motivation for this paper is to investigate the fixed point property for non-expansive mappings defined on $b$-fuzzy metric spaces. First, following the idea of S. Ješić's result from 2009, we introduce convex, strictly convex and normal structures for sets in $b$-fuzzy metric spaces. By using topological methods and these notions, we prove the existence of fixed points for self-mappings defined on $b$-fuzzy metric spaces satisfying a nonlinear type condition. This result generalizes and improves many previously known results, such as W. Takahashi's result on metric spaces from 1970. A representative example illustrating the main result is provided.