  We study Diophantine approximation in completions of functions fields over
finite fields, and in particular in fields of formal Laurent series over finite
fields. We introduce a Lagrange spectrum for the approximation by orbits of
quadratic irrationals under the modular group. We give nonarchimedean analogs
of various well known results in the real case: the closedness and boundedness
of the Lagrange spectrum, the existence of a Hall ray, as well as computations
of various Hurwitz constants. We use geometric methods of group actions on
Bruhat-Tits trees.
