Based on the notion of a computable p-adic number the notion of a polynomially time computable function over the field of p-adic numbers is introduced and studied. Theorems relating analytical properties with computability properties are established. The complexity of roots, and inverse function theorems are established at the level of polynomial time complexity. Relations between differentiability and polynomial time complexity and the maximization problem are discussed. Differences and similarities between the analogous questions for the real numbers are pointed out.