  A Banach space $E$ is said to be injective if for every Banach space $X$ and
every subspace $Y$ of $X$ every operator $t:Y\to E$ has an extension $T:X\to
E$. 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.
