The Fermi acceleration is always inherent in completely chaotic time-dependent billiards. At the same time, the particle dynamics in nearly integrable billiard systems can be more complex. Using a simplified approach, we investigate time-dependent stadium-like billiards and show that at a certain particle velocity, Vr, a resonance between external periodical perturbations and the motion within stability islands of the unperturbed billiard can be observed. This resonance suppresses the Fermi acceleration of particles with velocities less than Vr. As a result, we observe a separation of billiard particles by their velocities. If V0 < Vr, the average particle velocity decreases, while the particles with V0 > Vr are on average accelerated. At the initial velocity V0 = Vr we observe a phase transition in the velocity distribution of particles: if V0 < Vr then the distribution approaches a stationary one, whereas for V0 > Vr the distribution is non-stationary and spreads toward the higher velocities. This phenomenon may be treated as a peculiar billiard Maxwell's Demon, when weak perturbations of a system lead the particle ensemble to separation. In other nearly integrable billiard systems similar resonances can lead to differences in the acceleration of particles with velocities smaller or larger than a resonance value.