I apologize in advance if this is well-known.

Let $X$ be a Banach space. Let's call only for this post that $X$ is *self-injective* if for every closed subspaces
\begin{equation}
A\subseteq B\subseteq X \tag{*}\label{eq:1}
\end{equation}
and bounded linear $T:A\to X$, there exists an extension $\widetilde{T}:B\to X$. The difference from the definition of injectivity is the restriction \eqref{eq:1} that $A$, $B$ must be subspaces of $X$. We can define $\lambda$-self-injectivity similarly.

Clearly, injective and separably injective Banach spaces are self-injective in this sense.

Question:Are there other self-injective (especially $1$-self-injective) Banach spaces? Is there a characterization of self-injectivity?

**edit/update:** Thanks to Jesus Castillo's answer, the common terminology for these spaces is *extensible*. If I'm allowed, I'd like to be a voluntary advertiser of a group of interesting problems, which I'm eager to learn the answers. The snippet below is from the copy of the book (p.331-332) on books.google.com co-authored by Jesus Castillo.

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