Connective tissue displays a large compositional and structural complexity that involves multiple length scales. In particular, on the molecular and the nanometer level, the elementary processes that determine the biomechanics of collagen fibrils in connective tissues are still poorly understood. Here, we use atomic force microscopy (AFM) to determine the three-dimensional (3D) depth profiles of the local nanomechanical properties of collagen fibrils and their embedding interfibrillar matrix in native (unfixed), hydrated Achilles tendon of sheep and chickens. AFM imaging in air with controlled humidity preserves the tissue's water content, allowing the assembly of collagen fibrils to be imaged in high resolution beneath an approximately 5-10 nm thick layer of the fluid components of the interfibrillar matrix. We collect pointwise force-distance (FD) data and amplitude-phase-distance (APD) data, from which we construct 3D depth profiles of the local tip-sample interaction forces. The 3D images reveal the nanomechanical morphology of unfixed, hydrated collagen fibrils in native tendon with a 0.1 nm depth resolution and a 10 nm lateral resolution. We observe a diversity in the nanomechanical properties among individual collagen fibrils in their adhesive and in their repulsive, viscoelastic mechanical response as well as among the contact points between adjacent collagen fibrils. This sheds new light on the role of interfibrillar bonds and the mechanical properties of the interfibrillar matrix in the biomechanics of tendon.