  The following selection theorem is established:\\ Let $X$ be a compactum
possessing a binary normal subbase $\mathcal S$ for its closed subsets. Then
every set-valued $\mathcal S$-continuous map $\Phi\colon Z\to X$ with closed
$\mathcal S$-convex values, where $Z$ is an arbitrary space, has a continuous
single-valued selection. More generally, if $A\subset Z$ is closed and any map
from $A$ to $X$ is continuously extendable to a map from $Z$ to $X$, then every
selection for $\Phi|A$ can be extended to a selection for $\Phi$.
  This theorem implies that if $X$ is a $\kappa$-metrizable (resp.,
$\kappa$-metrizable and connected) compactum with a normal binary closed
subbase $\mathcal S$, then every open $\mathcal S$-convex surjection $f\colon
X\to Y$ is a zero-soft (resp., soft) map. Our results provide some
generalizations and specifications of Ivanov's results (see \cite{i1},
\cite{i2}, \cite{i3}) concerning superextensions of $\kappa$-metrizable
compacta.
