  We study the class of overconvergent subanalytic subsets of a $k$-affinoid
space $X$ when $k$ is a non-archimedean field. These are the images along the
projection $X \times B^n \to X$ of subsets defined with inequalities between
functions of $X\times B^n$ which are overconvergent in the variables of $B^n$.
In particular, we study the local nature, with respect to $X$, of
overconvergent subanalytic subsets. We show that they behave well with respect
to the Berkovich topology, but not to the $G$-topology. This gives
counter-examples to previous results on the subject, and a way to correct them.
Moreover, we study the case dim$(X)=2$, for which a simpler characterisation of
overconvergent subanalytic subsets is proven.
