Linear models with fixed effects and many dummy variables are common in some fields. Such models are straightforward to estimate unless the factors have too many levels. The R package lfe solves this problem by implementing a generalization of the within transformation to multiple factors, tailored for large problems.
Gaure, S. (2013). Lfe: Linear group fixed effects. R Journal, 5(2), 104–116. https://doi.org/10.32614/rj-2013-031