Serre's Renorming Problem

Let K be a non-archimedean complete valued field and let $(E,\|.\| )$ be a non-archimedean normed space over K. Let K× :=\ λ :λ ∈ K \ 0\ \ and E× :=\ x :x∈ E \0\ \. By Serre's Renorming Problem we mean the following question: can one for every non-archimedean normed space E introduce a norm \|.\| • on E that is equivalent to the given norm in the sense that it determines the same topology and has the property \|E×\|•= K×? An affirmative solution of this problem is presented.