  In this work, we study a continued fractions theory for the topological
completion of the field of Puiseux series. As usual, we prove that any element
in the completion can be developed as a unique continued fractions, whose
coefficients are polynomials in roots of the variable, and that this
approximation is the best ''rational'' Diophantine approximation of such
element. Then, we interpret the preceding result in terms of the action of a
suitable arithmetic subgroup of the special linear group on the Berkovich space
defined over the said completion. We also explore the connections between
points of type IV of the Berkovich space in terms of some ''non-convergent'' or
''undefined'' continued fractions, in a sense that we make precise in the text.
