A Banach space E is said to be injective if for every Banach space X and every subspace Y of X every operator t\colon Y\to E has an extension T\colon X \to E . We say that E is \aleph -injective (respectively, universally \aleph -injective) if the preceding condition holds for Banach spaces X (respectively Y ) with density less than a given uncountable cardinal \aleph . We perform a study of \aleph -injective and universally \aleph -injective Banach spaces which extends the basic case where \aleph=\aleph_1 is the first uncountable cardinal. When dealing with the corresponding "isometric" properties we arrive to our main examples: ultraproducts and spaces of type C(K) . We prove that ultraproducts built on countably incomplete \aleph -good ultrafilters are (1,\aleph) -injective as long as they are Lindenstrauss spaces. We characterize (1,\aleph) -injective C(K) spaces as those in which the compact K is an F_\aleph -space (disjoint open subsets which are the union of less than \aleph many closed sets have disjoint closures) and we uncover some projectiveness properties of F_\aleph -spaces.