  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 Alfv\'en 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 $k_\perp\rho_p>1$, 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.
