In their 1984 book [Algebraic combinatorics. I: Association schemes, Mathematics Lecture Note Series. Menlo Park, California etc.: The Benjamin/Cummings Publishing Company, Inc. Advanced Book Program. (1984; Zbl 0555.05019)], \textit{E. Bannai} and \textit{T. Ito} conjectured that there are only finitely many distance-regular graphs of fixed valency greater than two. Bannai and Ito showed that their conjecture holds for valencies \(k = 3, 4\), as well as for the special class of bipartite distance-regular graphs. J. H. Koolen and V. Moulton also showed that the conjecture holds for distance-regular graphs of fixed valency \(k = 5, 6\) or \(7\), and for triangle-free distance-regular graphs of fixed valency \(k = 8,\;9\) or \(10\). In this paper the authors show that the Bannai-Ito conjecture holds for regular near polygons and geodetic distance-regular graphs.