Using the \( \textrm{SO}\left(\mathcal{N}\right) \) superspace formulation for \( \mathcal{N} \) -extended conformal supergravity in three dimensions, we derive all maximally supersymmetric backgrounds in the \( \mathcal{N} \) = 4 case. The specific feature of this choice is that the so-called super Cotton tensor XIJKL = X[IJKL], which exists for \( \mathcal{N} \) ≥ 4, is equivalent to the scalar X defined by XIJKL = εIJKLX. This scalar may be used as a deformation parameter. In the family of (p, q) anti-de Sitter (AdS) superspaces with p + q = 4, p ≥ q, it is known that X ≠ 0 exists only if p = 4 and q = 0. In general, the (4, 0) AdS superspaces are characterised by the structure group SL(2, ℝ) × SO(4) and their geometry is determined by two constant parameters, S and X, of which the former determines the AdS curvature, while the R-symmetry curvature is determined by the parameters (X + 2S) and (X – 2S) in the left and right sectors of SU(2)L × SU(2)R, respectively. Setting S = 0 leads to the so-called deformed \( \mathcal{N} \) = 4 Minkowski superspace \( {\mathbbm{M}}_X^{\left.3\right|8} \) introduced thirteen years ago. We use projective-superspace techniques to construct general interacting supersymmetric field theories in \( {\mathbbm{M}}_X^{\left.3\right|8} \) and demonstrate that they originate as massive deformations of the following two families of \( \mathcal{N} \) = 4 theories in standard Minkowski superspace \( {\mathbbm{M}}^{\left.3\right|8} \) : (i) \( \mathcal{N} \) = 4 superconformal field theories; and (ii) \( \mathcal{N} \) = 4 supersymmetric gauge theories in \( {\mathbbm{M}}^{\left.3\right|8} \) which are not superconformal but possess the R-symmetry group SU(2)L × SU(2)R. Extensions of the theories in (ii) to \( {\mathbbm{M}}_X^{\left.3\right|8} \) necessarily contain Chern-Simons terms at the component level. We also demonstrate the generation of topologically massive \( \mathcal{N} \) = 4 supersymmetric gauge theories from radiative corrections in the hypermultiplet sector.
