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We define an analog of the Leja-Siciak-Zaharjuta subharmonic extremalfunction for a proper subset $E$ of the Berkovich projective line $P^1$ over afield with a non-archimedean absolute value, relative to a point $\zeta \not\in E$. When $E$ is a compact set with positive capacity, we prove that theupper semicontinuous regularization of this extremal function equals the Greenfunction of $E$ relative to $\zeta$. As a separate result, we prove theBrelot-Cartan principle, under the additional assumption that the Berkovichtopology is second countable.

An extremal subharmonic function in non-archimedean potential theory