  We prove a negative solution to the analogue of Hilbert's tenth problem for
rings of one variable non-Archimedean entire functions in any characteristic.
In the positive characteristic case we prove more: the ring of rational
integers is uniformly positive existentially interpretable in the class of
$\{0,1,t,+,\cdot,=\}$-structures consisting of positive characteristic rings of
entire functions on the variable $t$. From this we deduce uniform
undecidability results for the positive existential theory of such structures.
As a key intermediate step, we prove a rationality result for the solutions of
certain Pell equation (which a priori could be transcendental entire
functions).
