In this paper, we initiate the investigation of cosine families of bounded linear operators on non-Archimedean Banach spaces. We show some properties of non-Archimedean $C_{0}-$cosine operator functions. Examples are given to support our work and we will discuss the solvability of some homogeneous $p$-adic second-order differential equations where the parameter of $C_{0}$-cosine family of bounded linear operators belongs to a clopen ball $\Omega_{r}$ of the ground field $\mathbb{K}$.