On Banach envelopes and duals of Lipschitz-free $p$-spaces for $0<p<1$

With the aim to better understand the intricate geometry of the class of Lipschitz free $p$-spaces $\mathcal{F}_p(\mathcal{M})$ when $0<p<1$, in this note we study their Banach envelopes and prove that if $0<p<1$ and $ \mathcal{M}$ is a metric space then the Banach envelope map of $\mathcal{F}_p(\mathcal{M})$ is one-to-one, thus solving in the positive a problem raised by Kalton in [F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces, Israel J. Math. 170 (2009), 317-335]. This property has important applications to the linear structure of this family of spaces, being the most immediate one that the dual space of $ \mathcal{F}_p(\mathcal{M})$ separates the points of $\mathcal{F}_p(\mathcal{M})$.