Let K be a locally compact non-discrete field of characteristic \(p>2,\) and Q be a non-degenerate isotropic binary quadratic form with coefficients in K. We obtain asymptotic estimates for the number of solutions in the two-fold product of a discrete subring inside K, of the inequalities of the form \(|Q(x,y)|<\delta \) for some \(\delta >0,\) where \(| \cdot |\) is an ultrametric absolute value on K. The estimates are obtained in terms of continued fraction expansions of the coefficients of the quadratic form Q.
