  Let $X$ be a closed symplectic manifold equipped a Lagrangian torus fibration
over a base $Q$. A construction first considered by Kontsevich and Soibelman
produces from this data a rigid analytic space $Y$, which can be considered as
a variant of the $T$-dual introduced by Strominger, Yau, and Zaslow. We prove
that the Fukaya category of tautologically unobstructed graded Lagrangians in
$X$ embeds fully faithfully in the derived category of (twisted) coherent
sheaves on $Y$, under the technical assumption that $\pi_2(Q)$ vanishes (all
known examples satisfy this assumption). The main new tool is the construction
and computation of Floer cohomology groups of Lagrangian fibres equipped with
topological infinite rank local systems that correspond, under mirror symmetry,
to the affinoid rings introduced by Tate, equipped with their natural
topologies as Banach algebras.
