It is known that in the complex case the Fatou set of a polynomial has no wandering components. This is not true in the non-Archimedean case; there exist examples of polynomials having a wandering Fatou component. These examples are related to a phenomenon called wild ramification, a property that does not occur in the complex numbers. In this work, we will use combinatorics related to the Fibonacci numbers in order to obtain polynomials having a wandering Fatou component in the absence of the wild ramification phenomenon.