Keller and Ochsenius (1995) has a spectral theorem for finite-dimensional symmetric matrices over the field of formal power series with real coefficients $\mathbf{R}((t))$ (they actually have a more general result, but I'm just interested in this one for now), in the sense that every finite symmetric square matrix can be diagonalized by some orthogonal matrix with entries in the field.


Question: Is there a spectral theorem for symmetric infinite-dimensional matrices over this field?


What about the obvious generalization to Hermitian matrices and $\mathbf{C}$? (I believe the approach in the previous paper works for this case as well when the matrices are finite-dimensional, though they don't say it explicitly so I'm not completely confident)



spectral theorem for infinite-dimensional matrices
In fact, the field considered by Keller and Ochsenius is more complicated; it must have a Krull valuation with a specific kind of value group. They had several papers devoted to the infinite-dimensional case too:


H. Keller and H. Ochsenius, Spectral decompositions of operators on non-Archimedean orthomodular spaces. Int. J. Theor. Phys. 34, No.8, 1507-1517 (1995). 


H. Keller and H. Ochsenius, Bounded operators on non-archimedean orthomodular spaces. Math. Slovaca 45, No.4, 413-434 (1995).


H. Keller and H. Ochsenius, Residual spaces and operators on orthomodular spaces. In: Schikhof, W. H. (ed.) et al., p-Adic functional analysis. Proceedings of the fourth international conference, Nijmegen, Netherlands, June 3--7, 1996. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 192, 265-274 (1997). 


