We study the phase behavior of a classical system of particles interacting through a strictly convex soft-repulsive potential which, at variance with the pairwise softened repulsions considered so far in the literature, lacks a region of downward or zero curvature. Nonetheless, such interaction is characterized by two length scales, owing to the presence of a range of interparticle distances where the repulsive force increases, for decreasing distance, much more slowly than in the adjacent regions. We investigate, using extensive Monte Carlo simulations combined with accurate free-energy calculations, the phase diagram of the system under consideration. We find that the model exhibits a fluid-solid coexistence line with multiple re-entrant regions, an extremely rich solid polymorphism with solid-solid transitions, and waterlike anomalies. In spite of the isotropic nature of the interparticle potential, we find that among the crystal structures in which the system can exist, there are also a number of non-Bravais lattices, such as cI16 and diamond.