  Let $R$ be a finite valuation ring of order $q^r$ where $q$ is odd and $A$ be
a subset of $R$. In the present paper, we prove that there exists a point $u$
in the Cartesian product set $A\times A\subset R^2$ such that the size of the
pinned distance set at $u$ satisfies $$|\Delta_u(A\times A)|\gg \min\left\{q^r,
\frac{|A|^3}{q^{2r-1}}\right\}.$$ This implies that if $|A|\ge
q^{r-\frac{1}{3}}$, then the set $A\times A$ determines a positive proportion
of all possible distances.
