Just to be sure, according to the theory of General Relativity, my understanding is that mass curves space-time. My question is, can mass also curve space?
It's more sensible to talk about spacetime curvature than spatial curvature as the latter depends on how you foliate spacetime. For example de Sitter spacetime is not flat, but it can be foliated in ways that gives you flat spatial slices and in ways that give you curved spatial slices.
One way of seeing that gravity can never be described as purely spatial curvature is that whilst the path of a test particle in a gravitational field is independent of its mass, it is not independent of its speed.