Several L-functions with the names Dirichlet, Dedekind, Elliptic, and so on usually have p-adic counterparts, so called p-adic L-functions, which share many similar properties such as an evaluation formula at s=1, class number formula, and encryption of the arithmetic in analytic data. In the talk, we will propose another property of Dirichlet L-function to be interpreted p-adically. The property is algebraic-differential independence, which is a special case of universality of the function. We discuss how one can formulate the property in a p-adic way and present related results.