$p$-adic dimensions in symmetric tensor categories in characteristic $p$

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.