There was a problem about nearly injective metric spaces posed by Aronszajn and Panitchpakdi which I actually solved in the past but it still remains open (as long as I know) for the Banach spaces--so this is the ** QUESTION** for this thread. I feel that the Banach version is still more interesting.

Term *injective* in the context of the metric spaces and of the Banach spaces is equivalent to the A&P's term *hyperconvex*.

Let $\ m\ $ be a cardinal number. A metric space $\ (X\ d)\ $ is said to be $m$-hyperconvex $\ \Leftarrow:\Rightarrow\ $ for every $\ A\subseteq X\ $ such that $\ |A|<m,\ $ and for each $\ f:A\rightarrow \mathbb R\ $ satisfying

$$\forall_{x\ y\in A}\ \ f(x)+f(y) \ge d(x\ y)$$

there exists $\ a\in A\ $ for which $\ d(a\ x)\le f(x)\ $ for all $\ x\in A$.

Thus, to answer Aronszajn & Panitchpakdi's question in positive, for each inequality $\ 4\le m <\aleph_0\ $ there exits an $m$-hyperconvex space which is not an $(m+1)$-hyperconvex (also, $\ m=2\ $ is trivial, and $\ m=3\ $ is a *mood improver*). But

are there $m$-hyperconvex Banach spaces which are not $(m+1)$-hyperconvex? (Can they be finitely dimensional?).