Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.
As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the form $$p^n + a_1q + a_2q^2 + \cdots$$ By the Weierstrass preparation theorem, the residue fields of $\mathbb{Z}((x))$ are precisely fields of the form $\mathbb{Q}_p(\alpha)$, where $\alpha$ has positive $p$-adic valuation.
Another question is:
Does the Tate elliptic curve over Frac $\mathbb{Z}((x))$ satisfy weak weak approximation? (this would imply that Frac $\mathbb{Z}((x))$ is Hilbertian) Does it have the Hilbert property (does it satisfy Hilbert's irreducibility theorem?)