The Aleksandrov problem on non-Archimedean normed space

Let X and Y   be non-Archimedean normed spaces over a linear ordered non-Archimedean field KK with a non-Archimedean valuation. A mapping f : X → Y preserves distance n if for all x, y ∈ X with ∥x − y∥ = n it follows that ∥f(x) − f(y)∥ = n and conversely. In this paper we shall study, instead of isometries, mappings satisfying the weaker assumption that they preserve distance in both directions. We shall prove that such mappings are not very far from being isometries.