Deep learning models have shown encouraging capabilities for mapping accurately forests at medium resolution with TanDEM-X interferometric SAR data. Such models, as most of current state-of-the-art deep learning techniques in remote sensing, are trained in a fully-supervised way, which requires a large amount of labeled data for training and validation. In this work, our aim is to exploit the high-resolution capabilities of the TanDEM-X mission to map forests at 6 m. The goal is to overcome the intrinsic limitations posed by midresolution products, which affect, e.g., the detection of narrow roads within vegetated areas and the precise delineation of forested regions contours. To cope with the lack of extended reliable reference datasets at such a high resolution, we investigate self-supervised learning techniques for extracting highly informative representations from the input features, followed by a supervised training step with a significantly smaller number of reliable labels. A 1 m resolution forest/non-forest reference map over Pennsylvania, USA, allows for comparing different training approaches for the development of an effective forest mapping framework with limited labeled samples. We select the best-performing approach over this test region and apply it in a real-case forest mapping scenario over the Amazon rainforest, where only very few labeled data at high resolution are available. In this challenging scenario, the proposed self-supervised framework significantly enhances the classification accuracy with respect to fully-supervised methods, trained using the same amount of labeled data, representing an extremely promising starting point for large-scale, very high-resolution forest mapping with TanDEM-X data.