  We show that when a quantum many-body system is subjected to coherent
periodic driving, the response may exhibit exotic freezing behavior in high
driving frequency ($\omega$) regime. In a periodically driven classical
thermodynamic system, freezing at high $\omega$ occurs when $1/\omega$ is much
smaller than the characteristic relaxation time of the system, and hence the
freezing always increases there as $\omega$ is increased. Here, in the
contrary, we see surprising non-monotonic freezing behavior of the response
with $\omega$, showing curious peak-valley structure. Quite interestingly, the
entire system tends to freeze almost absolutely (the freezing peaks) when
driven with a certain combination of driving parameters values (amplitude and
$\omega$) due to coherent suppression of dynamics of the quantum many-body
modes, which has no classical analog. We demonstrate this new freezing
phenomenon analytically (supported by large-scale numerics) for a general class
of integrable quantum spin systems.
