In this article, we functorially associate definable sets to
$k$
-analytic curves, and definable maps to analytic morphisms between them, for a large class of
$k$
-analytic curves. Given a
$k$
-analytic curve
$X$
, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of
$X$
and show that they satisfy a bijective relation with the radial subsets of
$X$
. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of
$k$
-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser’s theorem on iso-definability of curves. However, our approach can also be applied to strictly
$k$
-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.