= Solution
Let $d=|f'(a_0)|>0$ and $q_0=|f(a_0)|/d^2<1$. At any step before reaching a root, set
$$
a_{i+1}=a_i+h_i,\qquad h_i=-\frac{f(a_i)}{f'(a_i)}.
$$
Assuming inductively that $|f'(a_i)|=d$ and $q_i=|f(a_i)|/d^2<1$, this gives
$$
\boxed{|a_{i+1}-a_i|=\frac{|f(a_i)|}{|f'(a_i)|}=dq_i.}
$$
All derivatives of a <polynomial> over $R$ are integral on $R$, so $d\le1$. Hence $|h_i|<1$, and the <ultrametric inequality> keeps $a_{i+1}$ in $R$. If a root is reached, retain that value for subsequent terms, and the asserted weak step inequality still holds.
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