A basic idea of the classical approach for obtaining necessary optimality conditions in optimal control is to construct suitable “needle-like control variations.” We use this idea to prove the main result of the present paper—a Pontryagin maximum principle for infinite-dimensional optimal control problems with pointwise terminal constraints in arbitrary Banach state space. By refining the classical variational technique we are able to replace the differentiability of the norm of the state space (guaranteed by the strict convexity of its dual norm, which is assumed in the known results) by a separation argument. We also drop another key assumption which is common in the existing literature on infinite-dimensional control problems—that the set of variations (in the state space) of the state trajectory's endpoint (resulting from the control variations) be finite-codimensional. Instead, we require only that it has nonempty interior in its closed affine hull. As an application of the abstract result we present an illustrative example—an optimal control problem for an age-structured system with pointwise terminal state constraints.