  We study the finite-dimensional continuous complex representations of
$\mathrm{SL}_2$ over the ring of integers of non-Archimedean local fields of
even residual characteristic. We prove that for characteristic two, the
abscissa of convergence of the representation zeta function is $1$, resolving
the last remaining open case of this problem. We additionally prove that,
contrary to the expectation, the group algebras of $\mathbb
C[\mathbb{SL}_2(\mathbb Z/(2^{2 r}))]$ and $\mathbb C[\mathbb{SL}_2(\mathbb
F_2[t]/(t^{2r}))]$ are not isomorphic for any $r > 1$. This is the first known
class of reductive groups over finite rings wherein the representation theory
in the equal and mixed characteristic settings is genuinely different. From our
methods, we explicitly obtain the primitive representation zeta polynomials of
$\mathrm{SL}_2\left (\mathbb F_2[t]/(t^{2r}) \right) $ and $\mathrm{SL}_2\left
(\mathbb Z/(2^{2r}) \right) $ for $1 \leq r \leq 3$.
