In this paper, we consider the orthogonal symplectic Lie superalgebra $\mathrm{o}\mathrm{s}\mathrm{p}(1,2)$ over an algebraically closed field of prime characteristic $p>2$. Using the classification of the simple modules of the Lie superalgebra $\mathrm{o}\mathrm{s}\mathrm{p}(1,2)$, we prove that every local superderivation of $\mathrm{o}\mathrm{s}\mathrm{p}(1,2)$ to any simple module is a superderivation.