Lipschitz free $$\varvec{p}$$-spaces for $$\varvec{0<p<1}$$ in the light of the Schur $$\varvec{p}$$-property and the compact reduction

Abstract The geometric analysis of non-locally convex quasi-Banach spaces presents rich and nuanced challenges. In this paper, we introduce the Schur p -property and the strong Schur p -property for $$0 &lt; p \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , providing new tools to deepen the understanding of these spaces, and the Lipschitz free p -spaces in particular. Moreover, by developing an adapted version of the compact reduction principle, we prove that Lipschitz free p -spaces over discrete metric spaces possess the approximation property, thereby answering positively a question raised by Albiac et al. in [4].