Is the residue field of an algebraically closed field with respect to a non-trivial valuation infinite?

Question

Let $K$ be an algebraically closed field with a non-trivial valuation. Let $\Bbb k$ be the residue field, i.e. $\Bbb k = R/\mathfrak m$ where $R$ is the valuation ring $R:=\{a\in K\mid \text{val} (a)\ge 0\}$ and $\mathfrak m$ is the maximal ideal. Is $\Bbb k$ infinite?

Answer 1

Yes, because $\Bbb k$ is algebraically closed as well. Consider a polynomial $P\in K[X]$. Then its roots necessarily have valuation zero.