  We develop a theory of optomechanical cooling with a squeezed input light
field. We show that Stokes heating transitions can be \emph{fully} suppressed
when the driving field is squeezed below the vacuum noise level at an
appropriately selected squeezing phase and for a finite amount of squeezing.
The quantum backaction limit to laser cooling can be therefore moved down to
zero and the resulting final temperature is then solely determined by the ratio
between the thermal phonon number and the optomechanical cooperativity
parameter, independently of the actual values of the cavity linewidth and
mechanical frequency. Therefore driving with a squeezed input field allows to
prepare nanomechanical resonators, even with low resonance frequency, in their
quantum ground state with a fidelity very close to one.
