  To every object $X$ of a symmetric tensor category over a field of
characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and
$\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension
$\text{dim}(X)$ of $X$, coinciding with the usual dimension when $X$ is a
vector space. We study properties of $\text{Dim}_{\pm}(X)$, and in particular
show that they don't always coincide with each other, and can take any value in
$\mathbb{Z}_p$. We also discuss the connection of $p$-adic dimensions with the
theory of $\lambda$-rings and Brauer characters.
