  Let $k$ be a non-archimedean complete field. We prove a substitute for the
reduced fiber theorem (of Bosch, L\"utkebohmert and Raynaud) that holds for
every morphism $Y\to X$ flat and with geometrically reduced fibers between
$k$-affinoid spaces in the sense of Berkovich, without assuming that $X$ and
$Y$ are strict, nor that the relative dimension of $Y$ over $X$ is constant. We
do not use the original reduced fiber theorem, nor the language or the
techniques of formal geometry. Our statement is formulated in terms of Temkin's
graded reduction; our proof rests on a finiteness result of Grauert and Remmert
and on Temkin's theory of (graded) reduction of germs of analytic spaces. It
will be used for describing the variation of the connected components of the
fiber of a quasi-smooth map in a forthcoming work on flattening in the
Berkovich setting.
