  Introduced by Angl\`{e}s, Pellarin, and Tavares Ribeiro, Drinfeld modules
over Tate algebras are closely connected to Anderson log-algebraicity
identities, Pellarin $L$-series, and Taelman class modules. In the present
paper we define the de Rham map for Drinfeld modules over Tate algebras, and we
prove that it is an isomorphism under natural hypotheses. As part of this
investigation we determine further criteria for the uniformizability and rigid
analytic triviality of Drinfeld modules over Tate algebras.
