On a theorem of Forelli in a non-standard setting

By a classical theorem of Forelli, a complex-valued function on the open unit ball in $\mathbb {C}^n$ is holomorphic if it is holomorphic along each vector line and satisfies some weak differentiability conditions near the origin. Given any uncountable real closed field $R$ (possibly non-Archimedean), we prove an analogous result for functions defined over the algebraic closure $C = R(\sqrt {-1})$ of $R$. We also give an example showing that uncountability of $R$ is essential.