The study of delay-fractional differential equations (fractional DEs) have recently attracted a lot of attention from scientists working on many different subjects dealing with mathematically modeling. In the study of fractional DEs the first question one might raise is whether the problem has a solution or not. Also, whether the problem is stable or not? In order to ensure the answer to these questions, we discuss the existence and uniqueness of solutions (EUS) and Hyers-Ulam stability (HUS) for our proposed problem, a nonlinear fractional DE with p-Laplacian operator and a non zero delay tau > 0 of order n - 1 < nu*, epsilon < n, for n >= 3 in Banach space A. We use the Caputo's definition for the fractional differential operators D-nu*, D-epsilon. The assumed fractional DE with p-Laplacian operator is more general and complex than that studied by Khan et al. Eur Phys J Plus, (2018);133:26.