We present a detailed study of the quasiparticle contribution to the low-temperature specific heat of an extreme type-II superconductor at high magnetic fields. Within a T-matrix approximation for the self-energies in the mixed state of a homogeneous superconductor, the electronic specific heat is a linear function of temperature with a linear-$T$ coefficient $gamma_s(H)$ being a nonlinear function of magnetic field $H$. In the range of magnetic fields $Hagt (0.15-0.2)H_{c2}$ where our theory is applicable, the calculated $gamma_s(H)$ closely resembles the experimental data for the borocarbide superconductor YNi$_2$B$_2$C.
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