  In recent work, the first two authors constructed a generalized continued
fraction called the $p$-continued fraction, characterized by the property that
its convergents (a subsequence of the regular convergents) are best
approximations with respect to the $L^p$ norm, where $p\geq 1$. We extend this
construction to the region $0<p<1$, where now the $L^p$ quasinorm is
non-convex. We prove that the approximation coefficients of the $p$-continued
fraction are bounded above by $1/\sqrt{5}+\varepsilon_p$, where
$\varepsilon_p\to 0$ as $p\to 0$. In light of Hurwitz's theorem, this upper
bound is sharp, in the limit. We also measure the maximum number of consecutive
regular convergents that are skipped by the $p$-continued fraction.
