Non-Archimedean holomorphic maps and the Ahlfors islands theorem

We present a p-adic and non-archimedean version of some classical complex holomorphic function theory. Our main result is an analogue of the Five Islands Theorem from Ahlfors' theory of covering surfaces. For non-archimedean holomorphic maps, our theorem requires only two islands, with explicit and nearly sharp constants, as opposed to the three islands without explicit constants in the complex holomorphic theory. We also present non-archimedean analogues of other results from the complex theory, including theorems of Koebe, Bloch, and Landau, with sharp constants.