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We consider the problem of counting polynomial curves on analytic ordefinable subsets over the field ${\mathbb{C}}(\!(t)\!)$, as a function of thedegree $r$. A result of this type could be expected by analogy with theclassical Pila-Wilkie counting theorem in the archimean situation.  Some non-archimedean analogs of this type have been developed in the work ofCluckers-Comte-Loeser for the field ${\mathbb{Q}}_p$, but the situation in${\mathbb{C}}(\!(t)\!)$ appears to be significantly different. We prove thatthe set of polynomial curves of a fixed degree $r$ on the transcendental partof a subanalytic set over ${\mathbb{C}}(\!(t)\!)$ is automatically finite, butgive examples showing that their number may grow arbitrarily quickly even foranalytic sets. Thus no analog of the Pila-Wilkie theorem can be expected tohold for general analytic sets. On the other hand we show that if one restrictsto varieties defined by Pfaffian or Noetherian functions, then the number growsat most polynomially in $r$, thus showing that the analog of Wilkie'sconjecture does hold in this context.

Point counting and Wilkie's conjecture for non-archimedean Pfaffian and Noetherian functions