Magnetic fluctuations in the solar wind are distributed according to Kolmogorov's power law f^-5/3 below the ion cyclotron frequency f_ci. Above this frequency, the observed steeper power law is usually interpreted in two different ways, as a dissipative range of the solar wind turbulence, or another turbulent cascade, the nature of which is still an open question. Using the Cluster magnetic data we show that after the spectral break the intermittency increases toward higher frequencies, indicating the presence of nonlinear interactions inherent to a new inertial range and not to the dissipative range. At the same time the level of compressible fluctuations rises. We show that the energy transfer rate and intermittency are sensitive to the level of compressibility of the magnetic fluctuations within the small-scale inertial range. We conjecture that the time needed to establish this inertial range is shorter than the eddy-turnover time, and is related to dispersive effects. A simple phenomenological model, based on the compressible Hall MHD, predicts the magnetic spectrum ~k^-7/3+2alpha, which depends on the degree of plasma compression alpha.