  Consider a rational map $R$ of degree $d\geq 2$ with coefficients over the
non-archimedean field $\mathbb{C}_p$, with $p$ a fixed prime number. If $R$ has
a cycle of Siegel disks and has good reduction, then it was shown by
Rivera-Letelier in his PhD dissertation 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 paper, we address the case of rational maps with bad reduction
and provide an extension of Rivera-Letelier's result for these class of maps.
