In this paper we introduce and study a new class of analytic and $p$-valent functions involving higher-order derivatives. For this $p$-valent function class, we derive several interesting properties including (for example) coefficient inequalities, distortion theorems, extreme points, and the radii of close-to-convexity, starlikeness and convexity. Several applications involving an integral operator are also considered. Finally, we obtain some results for the modified Hadamard product of the functions belonging to the $p$-valent function class which is introduced here.