  Given an ample line bundle $L$ over a projective $\mathbb{K}$-variety $X$,
with $\mathbb{K}$ a non-Archimedean field, we study limits of non-Archimedean
metrics on $L$ associated to submultiplicative sequences of norms on the graded
pieces of the section ring $R(X,L)$. We show that in a rather general case, the
corresponding asymptotic Fubini-Study operator yields a one-to-one
correspondence between equivalence classes of bounded graded norms and bounded
plurisubharmonic metrics that are regularizable from below. This generalizes
results of Boucksom-Jonsson where this problem has been studied in the
trivially valued case.
