gn.general topology - p-adic noninvariance of dimension

$mathbb{Q}_p$ is homeomorphic to a countable direct sum of copies of the Cantor set $C$. Indeed, because the valuation is discrete, for each $n geq 1$ the "annulus"

$A_n =$ {$x in mathbb{Q}_p | p^{n-1} < ||x|| leq p^{n}$}

is closed and homeomorphic to the Cantor set $C$. (Take of course $A_0 = mathbb{Z}_p$.)

Since as you observed above, $C times C cong C$, it follows that $mathbb{Q}_p^n$ and
$mathbb{Q}_p^m$ are homeomorphic for all $m, n in mathbb{Z}^+$ (and the homeomorphism type is independent of $p$).