  In this paper we deal with two weaker forms of injectivity which turn out to
have a rich structure behind: separable injectivity and universal separable
injectivity. We show several structural and stability properties of these
classes of Banach spaces. We provide natural examples of (universally)
separably injective spaces, including $\mathcal L_\infty$ ultraproducts built
over countably incomplete ultrafilters, in spite of the fact that these
ultraproducts are never injective. We obtain two fundamental characterizations
of universally separably injective spaces: a) A Banach space $E$ is universally
separably injective if and only if every separable subspace is contained in a
copy of $\ell_\infty$ inside $E$. b) A Banach space $E$ is universally
separably injective if and only if for every separable space $S$ one has
$\Ext(\ell_\infty/S, E)=0$. The final Section of the paper focuses on special
properties of 1-separably injective spaces. Lindenstrauss\ obtained in the
middle sixties a result that can be understood as a proof that, under the
continuum hypothesis, 1-separably injective spaces are 1-universally separably
injective; he left open the question in {\sf ZFC}. We construct a consistent
example of a Banach space of type $C(K)$ which is 1-separably injective but not
1-universally separably injective. We show that, under the continuum
hypothesis, "to be universally separably injective" is not a $3$-space
property, as we wrongly claimed in the paper mentioned in the title.
