You're right, sorry. The restriction to contractions is only needed to have $\ell_1(\mathbb R)$ as a countable coproduct of $I=\mathbb R$.

@YemonChoi: is it really closed ? The usual closed category of Banach is the one with contracting linear maps (which has countable coproducts).

This definition allows to define sequences in $X$ by recursion: given a first "element" $1\to X$ and a "recursion law" $X\to X$, it defines a sequence $N\to N$. Nicer properties are given by the symmetric closed structure, for instance, addition is given as a map $N\otimes N\to N$ which is defined by adjunction by a map $N\to \operatorname{Hom}(N,N)$, itself defined by the usual recursive definition in $\operatorname{\mathbf{Set}}$.

Actually, I would like a category where a Natural Number Object could exist. I have edited my question accordingly.

Thanks a lot! This is actually helpful for my original problem where $K$ was a $p$-adic field, and otherwise very interestingly gives an example with non-trivial extension. Do you have any idea of what this extension may or may not be in general (algebraic, finite, ...) ?