We consider an optimal control problem with Volterra-type integral equations on a nonfixed time interval subject to endpoint constraints, mixed state-control constraints of equality and inequality type, and pure state inequality constraints. The main assumption is the linear–positive independence of the gradients of active mixed constraints with respect to the control. We formulate first order necessary optimality conditions for an extended weak minimum, the notion of which is a natural generalization of the notion of weak minimum with account of variations of the time. The presented conditions generalize the local maximum principle in optimal control problems with ordinary differential equations.