<p>In this paper, we study the control problems of mean-field backward doubly stochastic differential equations with delay in the form of an integral with respect to a finite regular measure. Using the standard variational method, we introduce a new type of anticipated mean-field doubly stochastic differential equations as adjoint equations and derive a necessary condition in form of the maximum principle for optimal control. Under appropriate assumptions, the sufficiency of the maximum principle is also established. Our results can be applied to a certain class of linear quadratic control problems and be used to study the mean-field game for a pension fund model with delayed surplus.</p>