  Tate algebras are fundamental objects in the context of analytic geometry
over the p-adics. Roughly speaking, they play the same role as polynomial
algebras play in classical algebraic geometry. In the present article, we
develop the formalism of Gr{\"o}bner bases for Tate algebras. We prove an
analogue of the Buchberger criterion in our framework and design a
Buchberger-like and a F4-like algorithm for computing Gr{\"o}bner bases over
Tate algebras. An implementation in SM is also discussed.
