In this chapter we introduce the definition of a k-valued analytic function on an open set \(\Omega \) in \(k^n\) where k is a local field k and prove the main results about them. When \(k\simeq \mathbb {C}\) this is a familiar concept defined via the existence of the first derivative with respect to a complex variable. When \(k=\mathbb {R}\) one may define a \(\mathbb {R}\) -valued analytic function on \(\Omega \) as the restriction to \(\Omega \) of a \(\mathbb {C}\) -valued analytic function on an open set \(\tilde{\Omega }\) in \(\mathbb {C}^n\) containing \(\Omega \) . An alternative approach (following Weierstrass) is via convergent power series over \(\mathbb {R}\) . When k is not archimedean, only the second approach is available and is adopted in this chapter.