Playing fast and loose with size issues for simplicity, let $\mathfrak{S}$ be the structure of the surreal numbers equipped with addition, multiplication, and the simplicity order. I'm curious how much "$\mathbb{N}$-like arithmetic" we can (first-order, parameter-freely-)${}$definably locate within $\mathfrak{S}$. Note that without the simplicity order, the resulting structure would be too model-theoretically tame to interpret anything interesting from this perspective.
There are various ways to ask this. The most ambitious natural question here, in my opinion, is whether there is a definable subring of $\mathfrak{S}$ satisfying $\mathsf{PA}$. At first glance the subring Oz of omnific integers seem a plausible candidate, but in fact Oz is not even a model of the very weak subtheory $\mathsf{I}\Sigma_1$ of $\mathsf{PA}$; see here.
Based on this, it seems a good idea to look at a weaker theory of arithmetic first:
Q1: Is there a definable subring of $\mathfrak{S}$ satisfying $\mathsf{I\Sigma_1}$?
Q2 If not, does $Th(\mathfrak{S})$ at least interpret $\mathsf{I\Sigma_1}$?
The ordinals (with their ordering) can be interpreted inside $\mathfrak{S}$ as the equivalence classes with respect to the simplicity order. Inside the ordinals we can define the finite ordinals as those that are less than the first nonzero limit ordinal. So, we can define the set of surreal numbers with finite birthday, i.e. the dyadic rationals. A positive dyadic rational $x$ is an integer iff $x$ is simpler than $y$ for all dyadic rationals $y$ with $x-1<y<x$. So, $\mathbb{N}$ is definable in $\mathfrak{S}$.