Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an infinite model, i.e. an infinite field of characteristic $p$: $\mathbb{Z}_p(x)$, the field of quotients of the polynomial ring $\mathbb{Z}_p[x]$. This field is countably infinite. Since $T$ has an infinite model $T$ has models of arbitrarily large cardinality by the Upward Löwenheim Skolem theorem, i.e. there are fields of characteristic $p$ with arbitrarily large cardinality.
My question is, are there any explicit examples of uncountable fields of characteristic $p$?
Edit: As Eoin pointed out below one example can be obtained by considering fractions of polynomials over $\mathbb{Z}_p$ in uncountable many variables $x_i$, $i\in I$, $|I|>\omega$.
Can anyone think of any other examples?
A "natural" example is the field $k((x))$ of Laurent series over a field $k$ of characteristic $p$. (For example take $k = \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.) This is the fraction field of the integral domain $k[[x]]$ of formal power series over $k$. The ring of formal power series $k[[x]]$ is uncountable, as it bijects with the set of functions from $\mathbb{N}$ to $k$, which has cardinality $|k|^\omega \geq 2^\omega$. Thus its fraction field is uncountable.