We perform a numerical study of ghost condensation---in the so-called Overhauser channel---for $SU(2)$ lattice gauge theory in minimal Landau gauge. The off-diagonal components of the momentum-space ghost propagator ${G}^{cd}(p)$ are evaluated for lattice volumes $V={8}^{4}$, ${12}^{4}$, ${16}^{4}$, ${20}^{4}$, ${24}^{4}$ and for three values of the lattice coupling: $\ensuremath{\beta}=2.2$, $2.3$, $2.4$. Our data show that the quantity ${\ensuremath{\phi}}^{b}(p)={ϵ}^{bcd}{G}^{cd}(p)/2$ is zero within error bars, being characterized by very large statistical fluctuations. On the contrary, $|{\ensuremath{\phi}}^{b}(p)|$ has relatively small error bars and behaves at small momenta as ${L}^{\ensuremath{-}2}{p}^{\ensuremath{-}z}$, where $L$ is the lattice side in physical units and $z\ensuremath{\approx}4$. We argue that the large fluctuations for ${\ensuremath{\phi}}^{b}(p)$ come from spontaneous breaking of a global symmetry and are associated with ghost condensation. It may thus be necessary (in numerical simulations at finite volume) to consider $|{\ensuremath{\phi}}^{b}(p)|$ instead of ${\ensuremath{\phi}}^{b}(p)$, to avoid a null average due to tunneling between different broken vacua. Also, we show that ${\ensuremath{\phi}}^{b}(p)$ is proportional to the Fourier-transformed gluon field components ${\stackrel{\texttildelow{}}{A}}_{\ensuremath{\mu}}^{b}(q)$. This explains the ${L}^{\ensuremath{-}2}$ dependence of $|{\ensuremath{\phi}}^{b}(p)|$, as induced by the behavior of $|{\stackrel{\texttildelow{}}{A}}_{\ensuremath{\mu}}^{b}(q)|$. We fit our data for $|{\ensuremath{\phi}}^{b}(p)|$ to the theoretical prediction $(r/{L}^{2}+v)/({p}^{4}+{v}^{2})$, obtaining for the ghost condensate $v$ an upper bound of about $0.058\text{ }\text{ }{\mathrm{GeV}}^{2}$. In order to check if $v$ is nonzero in the continuum limit, one probably needs numerical simulations at much larger physical volumes than the ones we consider. As a by-product of our analysis, we perform a careful study of the color structure of the inverse Faddeev-Popov matrix in momentum space.