Gelfand's formula says that for a complex matrix $A \in \mathbb{C}^{n \times n}$, $$\rho(A) = \lim_{m \rightarrow \infty} \|A^m\|^{1/m},$$ where $\rho$ is the spectral radius (norm of maximal eigenvalue / maximal norm of eigenvalues) and $\|\cdot \|$ is any matrix norm.
I looked at the proof at http://en.wikipedia.org/wiki/Spectral_radius and to me it seems like this should be valid (maybe with slight changes to the proof) over any algebraically closed field $K = \overline{K}$ with a nontrivial valuation (in particular, over $\overline{\mathbb{Q}_p}$).
Can anyone confirm this?