  We prove an ultrametric q-difference version of the Maillet-Malgrange
theorem, on the Gevrey nature of formal solutions of nonlinear analytic
q-difference equations. Since \deg_q and \ord_q define two valuations on
{\mathbb C}(q), we obtain, in particular, a result on the growth of the degree
in q and the order at q of formal solutions of nonlinear q-difference
equations, when q is a parameter. We illustrate the main theorem by considering
two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation,
and a q-difference equation satisfied by the colored Jones polynomials of the
figure 8 knots, in the linear case.
  We consider also a q-analog of the Maillet-Malgrange theorem, both in the
complex and in the ultrametric setting, under the assumption that |q|=1 and a
classical diophantine condition.
