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We prove that the metric dimension of an affine plane of order $q\\geq13$ is $3q-4$ and describe all resolving sets of that size if $q\\geq 23$. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order $q\\geq 4$ is shown to fall between $2q-2$ and $3q-6$, while for Desarguesian biaffine planes the lower bound is improved to $8q/3-7$ under $q\\geq 7$, and to $3q-9\\sqrt{q}$ under certain stronger restrictions on $q$. 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