Valuation ring need not equal the integral closure over a noncomplete field

Give $\mathbb{Q}$ its $5$-adic valuation, take $L=\mathbb{Q}(i)$, and choose the extension associated with the prime $(2+i)$. Then $(2+i)/(2-i)$ lies in the chosen valuation ring but has negative valuation at the conjugate prime. It is therefore not in the integral closure of ${\mathbb{Z}}_{(5)}$ in $L$.