  This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine
spaces for large $t$, with a focus on the important index $\lambda=1$ case. We
prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a
block-transitive group of automorphisms that is of affine type. Moreover, we
show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs,
except possibly when the group is one-dimensional affine. Our approach involves
a consideration of the finite 2-homogeneous affine permutation groups.
