The representation theory of classical groups (such as $S_n$, $GL_n$, $O_n$, $Sp_{2n}$) depends on the rank parameter $n$. While this parameter is an integer, some important representation-theoretic quantities (such as dimensions of representations,
6j-symbols, etc.) are polynomial in $n$ for sufficiently large $n$. This allowed Deligne
and Milne to interpolate the representation categories of $GL_n(\mathbb{C})$ to arbitrary values $n\in \mathbb{C}$. Later Deligne generalized this to $S_n$, $O_n$, $Sp_{2n}$, leading to ``representation theory in complex rank".
This of course extends verbatim to any field of characteristic zero, but not to positive
characteristic, as the relevant polynomials in $n$ have rational coefficients with arbitrary denominators. However, these polynomials turn out to be integer-valued, i.e. they are
integral linear combinations of binomial coefficients $\binom{n}{k}$ for various $k$.
So while these polynomials don't extend to $n\in K$ for an algebraically closed field $K$
of characteristic $p$, by Lucas' theorem they are periodic in $n$ with some period $p^k$.
This opens the door to a $p$-adic interpolation of the representation categories
of classical groups, with $n\in \mathbb{Z}_p$. Such an interpolation was defined
by Deligne in his letter to Ostrik in March 2015. In this letter, Deligne defined
the tensor category ${\rm Rep}S_U$ over $K$, where $U$ is a nontrivial ultrafilter on positive integers, which interpolates the categories ${\rm Rep}S_n$, and conjectured that these categories depend only on the limit $\lim_{U}n=t\in \mathbb{Z}_p$ (i.e., we can denote this category by ${\rm Rep}S_t$). This was confirmed by Nate Harman for $p>3$.
I will discuss these developments, and also discuss the notion of p-adic dimension
of an object of a symmetric tensor category in characteristic $p$ which was introduced in my work with Harman and Ostrik. For example, in Deligne's category ${\rm Rep}S_t$
the p-adic dimension of the standard (permutation) representation is $t$.