Consider a rational map R of degree d>1 with coefficients over a non-archimedean field Cp, with p≥2 a fixed prime number. If R has a cycle of Siegel disks and has good reduction, then it was shown by Rivera-Letelier (2000) that a new rational map Q can be constructed from R, in such a way that Q will exhibit a cycle of m-Herman rings. In this talk, we address the case of rational maps with bad reduction and provide an extension of Rivera-Letelier's result for this case.