No, it's not possible, because of the generalized Reeb Stability Theorem, A generalization of the Reeb Stability Theorem, William P. Thurston, Topology, V13, pp 347--352, 1974. The theorem basically says that for any group of smooth diffeomorphisms of a manifold that has a fixed point where every element has first derivative trivial, the group action near that point has a generalized nilpotent structure --- the intersection of the lower central series is trivial, for some ordinal $\alpha$. (My original interest was for understanding holonomy around leaves of foliations.)

For a Lie group, this result implies that any action near a fixed point where it has $C^1$ contact to the identity factors through a nilpotent Lie group. For a Lie group with a compactly supported action, apply this to a point on the frontier of an orbit, to conclude that the orbit factors through the action of a nilpotent quotient (in particular the group modulo the smallest term in its lower central series). For the affine group, this quotient is $\mathbb R$.

No, it's not possible, because of the generalized Reeb Stability Theorem, A generalization of the Reeb Stability Theorem, William P. Thurston, Topology, V13, pp 347--352, 1974. The theorem basically says that for any group of $C^1$-smooth diffeomorphisms of a manifold that has a fixed point where every element has first derivative trivial, the group action near that point has a generalized nilpotent structure --- the intersection of the lower central series is trivial, for some ordinal $\alpha$. (My original interest was for understanding holonomy around leaves of foliations.)

This generalized nilpotence phenomenon is fairly obvious for anything detected by the Taylor expansion at a point: if you look at the Taylor series for a vector field, commutators of vector fields with trivial $0$th and first term vanish to an even higher order. The main point is to understanding diffeomorphisms (or vector fields) that either have $C^\infity$ contact to the identity, or are not smooth enough to analyze with a Taylor series. This phenomenon is also related to the phenomenon analyzed by Margulis and others, that discrete groups of Lie groups generated by "small" elements are nilpotent.

For a Lie group, this result implies that any action near a fixed point where it has $C^1$ contact to the identity factors through a nilpotent Lie group. For a Lie group with a compactly supported action, apply this to a point on the frontier of any orbit, to conclude that the orbit factors through the action of a nilpotent quotient (in particular the group modulo the smallest term in its lower central series). For the affine group, this quotient is $\mathbb R$.