  In the last two decades a number of structures on the classical space of
translation invariant valuations on convex bodies were discovered, e.g.
product, convolution, a Fourier type transform. In this paper a non-Archimedean
analogue of the space of such (even) valuations with similar structures is
constructed. It is shown that, like in the classical case, the new space
equipped with either product or convolution satisfies Poincar\'e duality and
hard Lefschetz theorem.
