Edit (8/21/24). The paper Integration on the Surreals by Ovidiu Costin and myself referred to below has now appeared online in Advances in Mathematics. It is a substantially revised and expanded version of the below referenced preprint that appeared on mathatXiv. A pdf file of the just-said version is available at https://doi.org/10.1016/j.aim.2024.109823.
In a comment on Timothy Chow's recent answer that cited a paper a by O. Costin, H. Friedman and myself concerned with integration on the surreals, I noted that the cited paper has now been superseded by a revised and expanded version of portions that paper, and that Costin and I expect to post the revised version on arXiv within the next few weeks. I'm pleased to note that the revision (Integration on the Surreals: https://arxiv.org/abs/2208.14331) has now appeared. A separate paper by Costin and Friedman concerned with a different portion of the original paper will appear on another occasion.