In this paper, using the techniques of value distribution theory in the $p$-adic case, we prove uniqueness theorems for nonconstant $p$-adic holomorphic curves and the existence of a class of strong uniqueness polynomials for algebraically nondegenerate $p$-adic holomorphic curves. It is shown that if $X$ is a hypersurface defined by a polynomial in this class, then $X$ is a unique range set for algebraically nondegenerate $p$-adic holomorphic curves.