= Solution
One strong form of the <Hensel lemma> is this: if a complete discretely valued field $K$, a polynomial $f\in\mathcal O_K[X]$, and $a\in\mathcal O_K$ satisfy
$$
v(f(a))>2v(f'(a)),
$$
then there is a unique root $\alpha$ in the ball $v(\alpha-a)>v(f'(a))$.
Set $a_{n+1}=a_n-f(a_n)/f'(a_n)$. Taylor expansion shows that the valuation of the error at least doubles at each step, while $v(f'(a_n))$ remains constant. Thus the corrections tend to zero geometrically, so completeness gives a limit $\alpha$. Continuity gives $f(\alpha)=0$. Applying the same Taylor estimate to two roots in the stated ball proves uniqueness. This is <Newton iteration over a valued field>.
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