Diffusion MRI provides a non-invasive probe of local fibre bundles and long-range anatomical connections to characterise the structural connectome. One way to achieve very high spatial resolution diffusion MRI data for connectivity investigations is to scan ex-vivo brains over many hours or days, ideally at ultra-high field strength to boost signal levels. However, conventional diffusion MRI acquisition techniques do not generally deliver good data quality for the challenging conditions of ex-vivo tissue, characterised by reduced diffusivities and relaxation times when compared to in vivo. In this work, we investigate the potential of the diffusion-weighted steady-state free precession (DW-SSFP) sequence for ex vivo diffusion imaging of the macaque brain using a 10.5 T human MRI scanner with a conventional ( G max = 70 mT/m ) gradient set. SNR-efficiency optimisations incorporating experimental relaxation times demonstrate that the DW-SSFP sequence is predicted to achieve improved or similar SNR efficiency compared to a diffusion-weighted spin- and stimulated-echo sequence. Importantly, DW-SSFP can achieve this with the additional benefit of negligible geometric distortions, unlike conventional diffusion MRI using an echo-planar imaging readout. Using optimised DW-SSFP sequence parameters, we propose a protocol at 0.4 mm isotropic resolution using a two-shell multi-orientation protocol (effective b-values of 3200 s/mm2 and 5600 s/mm2). We fit the data using Tensor, Ball and 3-Sticks and Constrained Spherical Deconvolution signal representations. The results demonstrate high-quality diffusivity estimates across the entire brain with the ability to resolve multiple fibre populations in challenging crossing-fibre regions. The data will be made fully open source and multimodal as part of the Center for Mesoscale Connectomics, providing a resource for future connectivity investigations.