Recently, the R2D2 paradigm, standing for ''Residual-to-Residual DNN series for high-Dynamic-range imaging'', was introduced for image formation in Radio Interferometry (RI) as a learned version of the traditional algorithm CLEAN. The first incarnations of R2D2 are limited to planar imaging on small fields of view, failing to meet the spherical-imaging requirement of modern telescopes observing wide fields. To address this limitation, we propose the spherical-imaging extension S-R2D2. Firstly, as R2D2, S-R2D2 encapsulates its minor cycles in existing 2D-Euclidean deep neural network (DNN) architectures, but adapts its iterative scheme to incorporate the wide-field measurement model mapping a spherical image to visibility data. We implemented this model as the composition of an efficient Fourier-based interpolator mapping the spherical image onto the equatorial plane, with the standard RI operator mapping the equatorial-plane image to visibility data. Importantly, the interpolation step must inevitably be performed at a lower-than-optimal resolution on the plane, to meet the high-resolution requirement on the sphere of wide-field imaging while preserving scalability. Therefore, secondly, we design S-R2D2's DNN training loss to jointly learn to correct the interpolation approximations and identify residual image structures on the sphere, ensuring consistency with the spherical ground truth using the adjoint plane-to-sphere interpolator. Finally, we demonstrate through simulations S-R2D2's capability to perform fast and accurate reconstructions of spherical monochromatic intensity images, across high-resolution, high-dynamic-range settings.