  For any compact metric space $M$, we prove that the locally flat Lipschitz
functions separate points (of $M$) uniformly if and only if $M$ is purely
1-unrectifiable, resolving a problem posed by Weaver in 1999. We subsequently
use this geometric characterization to answer several questions in Lipschitz
analysis. Notably, it follows that the Lipschitz-free space $\mathcal{F}(M)$
over a compact metric space $M$ is a dual space if and only if $M$ is purely
1-unrectifiable. Furthermore, for any complete metric space $M$, we deduce that
pure 1-unrectifiability actually characterizes some well-known Banach space
properties of $\mathcal{F}(M)$ such as the Radon-Nikod\'ym property and the
Schur property. A direct consequence is that any complete, purely
1-unrectifiable metric space isometrically embeds into a Banach space with the
Radon-Nikod\'ym property. Finally, we provide a solution to a problem of
Whitney from 1935 by finding a rectifiability-based characterization of
1-critical compact metric spaces, and we use this characterization to prove the
following: a bounded turning tree fails to be 1-critical if and only each of
its subarcs has $\sigma$-finite Hausdorff 1-measure.
