The book describes the definitions and properties of p-adic analytic functions in a complete algebraically closed ultrametric field. Various properties of p-adic exponential-polynomials are examined, such asthe Hermite-Lindemann theorem in a p-adic field, with a new proof. The order and type of growth for analytic functions arev studied , in tyhe whole field and in the complemental of an open disk, using the Motzkin meromorphic products. Thye Nevanlinna theory is widely explained, with various applications. Small functions are introduced with results of uniqueness. The question of whether the ring of analytic functions , in the whole field or in an open disk, is a Bezout ring. is also examined