A metric space $Y$ has the *binary intersection property* provided that whenever a collection of closed balls in $Y$ intersects pairwise, then there is a common intersection point.

Does the metric space $M$ of compact metric spaces under the Gromov-Hausdorff distance satisfy the binary intersection property?

The motivation is simple: I have a metric space $X$ with subspace $A$ and a Lipschitz map $f:A \to M$. I'd like to know if I can extend $f$ to all of $X$ without increasing the Lipschitz constant. It turns out (see Prop 1.4 here) that this binary intersection property is one of two hypotheses that must be satisfied by $M$ if it is to admit Lipschitz extensions for arbitrary metric space pairs $(X,A)$.