I'm curious about Faltings' ["A $p$-adic Simpson correspondence"][1]. Do you know more detailed, introductory,  expositions, surveys, texts of seminars on that?

Edit: Annette Werner's survey ["Vector Bundles on Curves over $\mathbb{C}_p$"][2] seems to be related.

Edit: The [first part][3] of a "new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0". [Another related article][4].

Edit: today new in arXiv - ["Non-abelian Hodge theory for algebraic curves over characteristic p"][5]

  [1]: https://doi.org/10.1016/j.aim.2005.05.026 "Adv. Math. 198, No. 2, 847-862 (2005). zbMATH review at https://zbmath.org/1102.14022"
  [2]: https://www.uni-frankfurt.de/50581177/survey.pdf "Weng, Lin (ed.) et al., Arithmetic geometry and number theory. Hackensack, NJ: World Scientific. Series on Number Theory and its Applications 1, 47-64 (2006), doi:10.1142/9789812773531_0003. zbMATH review at https://zbmath.org/1110.14030"
  [3]: https://arxiv.org/abs/1102.5466 "Ahmed Abbes, Michel Gros. Sur la correspondance de Simpson p-adique. I : étude locale. Published at doi:10.1515/9781400881239. zbMATH review at https://zbmath.org/1342.14045"
  [4]: https://hal.science/file/index/docid/337672/filename/Higgs.pdf "Gros, Michel; Le Stum, Bernard; Quirós, Adolfo. A Simpson correspondence in positive characteristic. Publ. Res. Inst. Math. Sci. 46, No. 1, 1-35 (2010). zbMATH review at https://zbmath.org/1200.14036"
  [5]: https://arxiv.org/abs/1306.0299 "Tsao-Hsien Chen, Xinwen Zhu. Non-abelian Hodge theory for algebraic curves in characteristic p. Published at doi:10.1007/s00039-015-0343-6. zbMATH review at https://zbmath.org/1330.14015"