  Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of
an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let
$f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$,
where the coefficients are meromorphic functions defined at all parameters $c$
in some open disk $D\subseteq\mathbb{C}_p$. Assuming an appropriate stability
condition, we prove that the parameters $c$ for which $f_c$ is postcritically
finite (PCF) are isolated from one another in the $p$-adic disk $D$, except in
certain trivial cases. In particular, all PCF parameters of the family
$f_c(z)=z^d+c$ are $p$-adically isolated.
