We propose an adaptive, smooth, non-convex and sparsity-promoting variational model for singleimage super-resolution of real murine Optical Coherence Tomography (OCT) data. We follow a sparse-representation approach where sparsity is modelled with respect to a suitable dictionary generated from high-resolution OCT data. To do so, we employ pre-learned dictionaries tailored to model α-stable statistics in the non-Gaussian case, i.e. α < 2. The image reconstruction problem renders here particularly challenging due to the high level of noise degradation and to the heterogeneity of the data at hand. As a regulariser, we employ a smooth, non-convex and separable Cauchy-type penalty. To favour adaptivity to heterogeneous image contents, we propose a space-variant modelling by which the local degree of non-convexity encoded patch-wise within the local Cauchy shape parameter is estimated via maximum likelihood. For the solution of the patch-based smooth non-convex optimisation problems, we consider an extension of the cautious Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm where the descent direction is suitably updated depending on the local convexity behaviour of the functional. Our numerical results show that the combination of a space-variant modelling with a fast optimisation strategy improves reconstruction results, maintaining tissue texture and suppressing background noise to a desirable amount at the same time. Furthermore, the proposed optimisation strategy significantly reduces the computational efforts which often represent a major limitation in the analysis of OCT data.