When I look to the Norm article of Wikipedia, it seems that a norm is defined on a vector space over a subfield of the complex. In France, while this is usually the case, the definition of a vector space is a bit more general, and a vector space can be over a field endowed with an Absolute value.
Is there a name in English for those "Normed vector spaces"?
The Wikipedia article you cite is written from the perspective of an analyst, so it takes a narrow interpretation of a norm. A search for "normed linear/abstract space" together with "absolute value" or "valued field" shows that this term is also used for linear spaces over fields other than subfields of $\mathbb C$. Here is one example and here is another, with a screenshot of the relevant part: