Understanding the nature of the turbulent fluctuations below the ion gyroradius in solar-wind turbulence is a great challenge. Recent studies have been mostly in favor of kinetic Alfven wave (KAW) type of fluctuations, but other kinds of fluctuations with characteristics typical of magnetosonic, whistler and ion Bernstein modes, could also play a role depending on the plasma parameters. Here we investigate the properties of the sub-proton-scale cascade with high-resolution hybrid-kinetic simulations of freely-decaying turbulence in 3D3V phase space, including electron inertia effects. Two proton plasma beta are explored: the intermediate $beta_p=1$ and low $beta_p=0.2$ regimes, both typically observed in solar wind and corona. The magnetic energy spectum exhibits $k_perp^{-8/3}$ and $k_|^{-7/2}$ power laws at $beta_p=1$, while they are slightly steeper at $beta_p=0.2$. Nevertheless, both regimes develop a spectral anisotropy consistent with $k_|sim k_perp^{2/3}$ at $k_perprho_p>1$, and pronounced small-scale intermittency. In this context, we find that the kinetic-scale cascade is dominated by KAW-like fluctuations at $beta_p=1$, whereas the low-$beta$ case presents a more complex scenario suggesting the simultaneous presence of different types of fluctuations. In both regimes, however, a non-negligible role of ion Bernstein type of fluctuations at the smallest scales seems to emerge.
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