In the ring $\mathbb{Z}_p$ of $p$-adic integers it is quite easy to generalise the concept of divisibility by the prime $p$ to numbers that are not actual integers or rationals themselves. Just check whether the $p$-adic expansion of a $p$-adic integer $x$ ends with a $0$; if it does, call $x$ divisible by $p$. For example, the 2-adic integer $(1+\sqrt{-7})=...0010110110_2$ is divisible by 2 in that sense.
Now, it is also easy to see whether a member of $\mathbb{Z}_p$ that is an actual integer or a rational number, is divisible by another prime $q≠p$. Just calculate its representation in $\mathbb{Q}$, and if its $q$-valuation is strictly positive, call it divisible by $q$.
But is there a meaningful way of saying that a $p$-adic integer that is not itself a rational number is (or is not) divisible by another prime number $q$?
So this is always true. That's because if $q \neq p$ is prime, then $q$ is a unit in the ring $\mathbb{Z}_p$, and if we want to define $a/q$ for $a$ some arbitrary element of $\mathbb{Z}_p$, we can define it to be $aq^{-1}$.
To compute $q^{-1}$, note that you can compute it mod $p^n$ for any $n$, and take a limit.