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This article extends recent results on log-Coulomb gases in a $p$-field $K$(i.e., a nonarchimedean local field) to those in its projective line$\mathbb{P}^1(K)$, where the latter is endowed with the $PGL_2$-invariant Borelprobability measure and spherical metric. Our first main result is an explicitcombinatorial formula for the canonical partition function of log-Coulomb gasesin $\mathbb{P}^1(K)$ with arbitrary charge values. Our second main result iscalled the "$(q+1)$th Power Law", which relates the grand canonical partitionfunctions for one-component gases in $\mathbb{P}^1(K)$ (where all particleshave charge 1) to those in the open and closed unit balls of $K$ in a simpleway. The final result is a quadratic recurrence for the canonical partitionfunctions for one-component gases in both unit balls of $K$ and in$\mathbb{P}^1(K)$. In addition to efficient computation of the canonicalpartition functions, the recurrence provides their "$q\to 1$" limits and"$q\mapsto q^{-1}$" functional equations.

Log-Coulomb gases in the projective line of a $p$-field