  In 1959, Arens and Eells proved that every metric space can be isometrically
embedded into a real linear space as a closed subset. In later years, Michael
pointed out that every metric space can be isometrically embedded into a real
linear space as a linear independent subset and provided a short proof of
Arens--Eells theorem as an application. In this paper, we prove a
non-Archimedean analogue of the Arens--Eells isometric embedding theorem, which
states that for every non-Archimedean valued field $K$, every ultrametric space
can be isometrically embedded into a non-Archimedean valued field that is a
valued field extension of $K$ such that the image of the embedding is
algebraically independent over $K$. Using Levi-Civita fields, we also show that
every Urysohn universal ultrametric sapce has a valued field structure.
