TITLE: Discrete non archimedean valued field with infinite residue field
QUESTION [6 upvotes]: Let $\mathbb{K}$ a field with a discrete non archimedean absolute value and $A$ its ring of integers. Then $A$ is Noetherian, every ideal is generated by a power of $\pi\in A$, a uniformizer, and its unique maximal ideal is $\mathfrak{M}=\pi A$.
For instance, $\mathbb{Q}_p$ is such a field. Its residue field is $A/\mathfrak{M}\cong\mathbb{Z}/p\mathbb{Z}=\mathbb{F}_p$, the finite field of $p$ elements. 
Is there any field $\mathbb{K}$ whose residue field is not finite?

REPLY [2 votes]: You can take the ring of formal power series $K[[X]]$  over an infinite field $K$. It is a local ring, with maximal ideal generated by $X$, so that its residue field is (isomorphic to) $K$. The valuation of a power series is its order and it's easy to check it is non-archimedean.