On simultaneous uniform approximation to a $p$-adic number and its square

Let $p$ be a prime number. We show that a result of Teulié is nearly best possible by constructing a $p$-adic number $\xi$ such that $\xi$ and $\xi^2$ are uniformly simultaneously very well approximable by rational numbers with the same denominator. The same conclusion was previously reached by Zelo in his PhD thesis, but our approach using $p$-adic continued fractions is more direct and simpler.