We deal with polynomial ordinary differential equations of arbitrary order over valued fields, and more generally, the solution of twisted polynomials with coefficients in an ultrametric semi-normed ring. We treat systems of equations. The valued fields are not assumed archimedean.
The method is a far-reaching elaboration of Hensel ´ s lemma, which is replaced by the fixed point theorem, when the space is spherically complete, leading to solutions. If required, the space is extended to a spherical completion, and an extension theorem, reminiscent of the Hahn-Banach theorem, then leads to asymptotic approximations to solutions. Examples will be given in the case when the coefficients are germs of functions in some Hardy field, endowed with its natural (non-archimedean) valuation.