  Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic
space in the sense of Berkovich. In this note, we prove the equivalence between
three properties: 1) for every complete valued extension $k'$ of $k$, every
coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of
complex geometry (holomorphically separated, holomorphically convex) and higher
cohomology groups of the structure sheaf vanish (this latter hypothesis is
crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion
by compact analytic domains considered by Liu in his counter-example to the
cohomological criterion for affinoidicity.
  When $X$ has no boundary the characterization is simpler: in~2) the vanishing
of higher cohomology groups of the structure sheaf is no longer needed, so that
we recover the usual notion of Stein space in complex geometry; in 3) the
domains considered by Liu can be replaced by affinoid domains, which leads us
back to Kiehl's definition of Stein space.
  v2: major revision to handle also the case of spaces with boundary

Algebraic Geometry