This is the problem 2.11 from Lehman book "Theory of point estimation" 2-nd edition.
Construct a sequence $\{\delta_{n}\}$ of estimators of $g(\theta)$, satisfying
$$ \sqrt{n}[\delta_{n} - g(\theta)]\stackrel{d}{\to}\mathcal{N}[0,v(\theta)], \; v>0, $$
but for which the bias $b_{n}(\theta) = E[\delta_{n}] - g(\theta)$ does not tend to zero.
By another words, asymptotic normality does not guaranty that $\{\delta_{n}\}$ is unbiased or even that its bias tends to zero (p 439).