The recent paradigm shift introduced by the so-called (model-free) Data-Driven Computational Mechanics (DDCM) has led to a range of promising applications, including the acceleration of FE$^2$ simulations—particularly in the context of infinitesimal strain nonlinear elasticity. The key to this improvement lies in the online, simulation-specific enrichment of the material database via computational homogenization, which significantly reduces the number of microscale solver calls required compared to conventional FE2 methods. In contrast to other data-driven techniques, such as neural network-based surrogate models, DDCM has demonstrated excellent performance when provided with sufficiently rich datasets, while requiring negligible training time (just the initialisation of tree-based nearest-neighbor search data structures). However, it tends to yield inaccurate results when the data is sparse. Therefore, to ensure the efficiency of an active DDCM framework, it is essential to incorporate reliable error estimators that guide targeted database enrichment, along with modified DDCM solvers capable of effectively handling sparse data scenarios. Although conceptually straightforward at first sight, extending this methodology to the finite strain regime introduces additional computational challenges. DDCM in this setting is generally less efficient due to the increased nonlinearity of the resulting problems. Nevertheless, since FE2 simulations under finite strain are intrinsically more computationally demanding, there is greater potential for reducing overall costs—even if the relative speed-up is lower than in the infinitesimal strain case. The primary objective of this presentation is to evaluate the efficiency and computational benefits of the active DDCM approach in the finite strain setting, supported by a series of representative numerical examples—including 3D simulations—and an analysis of the resulting speed-ups. Implementation aspects will also be addressed within the framework of ddfenics, an open-source DDCM platform built on FEniCSx. In particular, recent advancements involving GPU-accelerated nearest-neighbour searches will be presented.