An Ahlfors Islands Theorem for non-archimedean meromorphic functions

We present a p-adic and non-archimdean version of the Five Islands Theorem for meromorphic functions from Ahlfors' theory of covering surfaces. In the non-archimedean setting, the theorem requires only four islands, with explicit constants. We present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed. This paper extends an earlier theorem of the author for holomorphic functions.