  We consider the differential KZ equations over $\mathbb C$ in the case, when
the hypergeometric solutions are one-dimensional hyperelliptic integrals of
genus $g$. In this case the space of solutions of the differential KZ equations
is a $2g$-dimensional complex vector space.
  We also consider the same differential equations modulo $p^s$, where $p$ is
an odd prime number and $s$ is a positive integer, and over the field $\mathbb
Q_p$ of $p$-adic numbers.
  We describe a construction of polynomial solutions of the differential KZ
equations modulo $p^s$. These polynomial solutions have integer coefficients
and are $p^s$-analogs of the hyperelliptic integrals. We call them the
$p^s$-hypergeometric solutions. We consider the space $\mathcal M_{p^s}$ of all
$p^s$-hypergeometric solutioins, which is a module over the ring of polynomial
quasi-constants modulo $p^s$. We study basic properties of $\mathcal M_{p^s}$,
in particular its natural filtration, and the dependence of $\mathcal M_{p^s}$
on $s$.
  We show that as $s$ tends to infinity, the limit of $\mathcal M_{p^s}$
defines a $g$-dimensional vector space of solutions of the differential KZ
equations over the field $\mathbb Q_p$.
