  We develop the theory of $p$-adic confluence of $q$-difference equations. The
main result is the surprising fact that, in the $p$-adic framework, a function
is solution of a differential equation if and only if it is solution of a
$q$-difference equation. This fact implies an equivalence, called
``Confluence'', between the category of differential equations and those of
$q$-difference equations. We obtain this result by introducing a category of
``sheaves'' on the disk $\mathrm{D}^-(1,1)$, whose stalk at 1 is a differential
equation, the stalk at $q$ is a $q$-difference equation if $q$ is not a root of
unity $\xi$, and the stalk at a root of unity is a mixed object, formed by a
differential equation and an action of $\sigma_\xi$.
