In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In particular,
we study the geometry at infinity of such Ricci solitons under the assumption that their tangent flow at infinity is the product of $mathbb{R}$ with a $3$-dimensional spherical space form. We also classify the tangent flows at infinity of $4$-dimensional steady soliton singularity models in general.
