Imagine that $S$ is an open subset of some fine moduli space which does not contain any rational curves. This does happen although I don't know any examples. Then any family over an affine space will have to be constant, so the answer would be no.
On the other hand, of course there is formal family over the completion of $S$ at the given point $p$ ($\hat{S}_p = \lim Spec (\mathcal{O}_{S, p}/\mathfrak{m}_{S, p}^n)$), which looks like the infinitesimal neighborhood of $0$ in the tangent space ($\hat{T}_0 = Spf(k[[t_1, \ldots, t_s]])$ where the $t_i$ form a basis of the dual of $T= T_{p} S$). I think morally this might play the role of what you want.