Giles Gardam's paper [A counterexample to the unit conjecture for group rings](https://doi.org/10.4007/annals.2021.194.3.9) was a big breakthrough in group rings. Giles produces a group $G$ that is torsionfree, such that there are units in $\mathbb{F}_2[G]$ that are not of the trivial form "a nonzero constant from $\mathbb{F}_2$ times an element of $G$".  He explicitly names a couple of these nontrivial units.


  [1]: https://projecteuclid.org/journals/annals-of-mathematics/volume-194/issue-3/A-counterexample-to-the-unit-conjecture-for-group-rings/10.4007/annals.2021.194.3.9.full