For the record, I deleted an answer as I misunderstood the question.

Find a configuration $(18_4)$ in [Bokowski, J., & Schewe, L. (2013). On the finite set of missing geometric configurations (n4). Computational Geometry, 46(5), 532-540]. It is nicely realized with triangles and infinity points, might be useful for the construction here.

Does the existence of such a non-trivial arrangement imply a $(4,4)$-configuration in the Euclidean plane? I know Grünbaum-RIgby configuration with this property.

@MoisheKohan I mean neighborhood of a point.

@TimothyBudd Yes, it's indeed very similar. That's why I believe there must already be some studies. I am only aware of minimal surface theorists that desingularize $\mathbb{R}\times G$ into minimal surfaces (not surprising).