  We discuss the Bethe ansatz in the Gaudin model on the tensor product of
finite-dimensional $sl_2$-modules over the field $F_p$ with $p$ elements, where
$p$ is a prime number. We define the Bethe ansatz equations and show that if
$(t^0_1,\dots,t^0_k)$ is a solution of the Bethe ansatz equations, then the
corresponding Bethe vector is an eigenvector of the Gaudin Hamiltonians. We
characterize solutions $(t^0_1,\dots,t^0_k)$ of the Bethe ansatz equations as
certain two-dimensional subspaces of the space of polynomials $F_p[x]$. We
consider the case when the number of parameters $k$ equals 1. In that case we
show that the Bethe algebra, generated by the Gaudin Hamiltonians, is
isomorphic to the algebra of functions on the scheme defined by the Bethe
ansatz equation. If $k=1$ and in addition the tensor product is the product of
vector representations, then the Bethe algebra is also isomorphic to the
algebra of functions on the fiber of a suitable Wronski map.
