The following theorem is proven in Jensen-Lenzig's "Algebra, Logic, and Applications, vol 2" Sublemma 3.34.1:
Suppose that $R$ is a regular local Henselian ring with maximal ideal $\mathfrak{m}$ and $R/\mathfrak{m}$ is quadratically closed and not of characteristic two. Then the equation $X^4 + Y^4 = Z^2$ has no pairwise relatively prime solutions in $\mathfrak{m}$.
They use this to show that the fraction field of $R$ defines $R$ (in the first order language of fields). I've also used this lemma in a crucial way for something.
My question is if there is a known broader context that this fits into. Jensen and Lenzig don't pursue the matter any further, they just need it as a lemma, so I would be very curious to know if this has been pursued by someone else. Obviously it's an analogue of a well-known theorem of Fermat. Is there anything known about analogues of Fermat's last theorem over regular local rings? (I've seen versions of FLT for function fields, but never for formal power series fields.) I would also like to know if there is a geometric proof.
