We study static, spherically symmetric solutions in a recently proposed ghost-free model of nonlinear massive gravity. We focus on a branch of solutions where the helicity-0 mode can be strongly coupled within certain radial regions, giving rise to the Vainshtein effect. We truncate the analysis to scales below the gravitational Compton wavelength, and consider the weak field limit for the gravitational potentials, while keeping all nonlinearities of the helicity-0 mode. We determine analytically the number and properties of local solutions that exist asymptotically on large scales, and of local (inner) solutions that exist on small scales. We find two kinds of asymptotic solutions, one of which is asymptotically flat, while the other one is not, and also two types of inner solutions, one of which displays the Vainshtein mechanism, while the other exhibits a self-shielding behavior of the gravitational field. We analyze in detail in which cases the solutions match in an intermediate region. The asymptotically flat solutions connect only to inner configurations displaying the Vainshtein mechanism, while the nonasymptotically flat solutions can connect with both kinds of inner solutions. We show furthermore that there are some regions in the parameter space where global solutions do not exist, and characterize precisely in which regions of the phase space the Vainshtein mechanism takes place.