Kalton's question: does the Lipschitz-free space of every uniformly discrete metric space have the bounded approximation property?

Theorem 1.1: there is a countable, unbounded, nonproper pointed metric space with $d(x,y)\ge1$ for distinct points whose real Lipschitz-free space has AP but fails $\lambda$-BAP for every finite $\lambda$. It is built as a wedge of bounded blocks (Theorem 1.2) in which a finite anchor set defeats every finite-rank operator of norm at most $p$. Consequences stated in the paper: for each $p$, real $\ell_1$ has an equivalent norm with $2B_p$-BAP failing $p$-BAP (for $p=1$ failing MAP), and $\mathcal F(M)$ is a separable Banach space that is not approximable in Kalton's nonlinear sense (Kalton 2012, Problem 1). Real scalars only; the proper countable case is excluded by Dalet's theorem.

Theorem 1.1: there is a countable, unbounded, nonproper pointed metric space with $d(x,y)\ge1$ for distinct points whose real Lipschitz-free space has AP but fails $\lambda$-BAP for every finite $\lambda$. It is built as a wedge of bounded blocks (Theorem 1.2) in which a finite anchor set defeats every finite-rank operator of norm at most $p$. Consequences stated in the paper: for each $p$, real $\ell_1$ has an equivalent norm with $2B_p$-BAP failing $p$-BAP (for $p=1$ failing MAP), and $\mathcal F(M)$ is a separable Banach space that is not approximable in Kalton's nonlinear sense (Kalton 2012, Problem 1). Real scalars only; the proper countable case is excluded by Dalet's theorem.