In this paper we present NR-SLAM, a novel non-rigid monocular SLAM system founded on the combination of a Dynamic Deformation Graph with a Visco-Elastic deformation model. The former enables our system to represent the dynamics of the deforming environment as the camera explores, while the later allows us to model general deformations in a simple way. The presented system is able to automatically initialize and extend a map modeled by a sparse point cloud in deforming environments, that is refined with a sliding-window Deformable Bundle Adjustment. This map serves as base for the estimation of the camera motion and deformation and enables us to represent arbitrary surface topologies, overcoming the limitations of previous methods. To assess the performance of our system in challenging deforming scenarios, we evaluate it in several representative medical datasets. In our experiments, NR-SLAM outperforms previous deformable SLAM systems, achieving millimeter reconstruction accuracy and bringing automated medical intervention closer. For the benefit of the community, we make the source code public.