A new class of functions is studied. We define the Brauer-Schur functions $B^{(p)}_{\lambda}$ for a prime number $p$, and investigate their properties. We construct a basis for the space of symmetric functions, which consists of products of $p$-Brauer-Schur functions and Schur functions. We will see that the transition matrix from the natural Schur function basis has some interesting numerical properties.