Colloidal supraparticles are finite, spherical assemblies of many primary particles. To take advantage of their emergent functionalities, such supraparticles must retain their structural integrity. Here, we investigate their size-dependent mechanical properties via nanoindentation. We find that the deformation resistance inversely scales with the primary particle diameter, while the work of deformation is dependent on the supraparticle diameter. We adopt the Griffith theory to such particulate systems to provide a predictive scaling to relate the fracture stress to the geometry of supraparticles. The interplay between primary particle material and cohesive interparticle forces dictates the mechanical properties of supraparticles. We find that enhanced stability, associated with ductile fracture, can be achieved if supraparticles are engineered to dissipate more energy via deformation of primary particles than breaking of interparticle bonds. Our work provides a coherent framework to analyze, predict, and design the mechanical properties of colloidal supraparticles.