Hiding in some secret compartment while soldiers scour the house, covering the baby's mouth to prevent it from inadvertently crying out, and smothering it to death lest the entire group be revealed is certainly a trope in the popular memory of the Holocaust, but it is one which is absolutely based on truth. We can't say at all how extensive it was - given the death rate of the Holocaust, any number of cases where it happened, but in the end were unsuccessful in preventing capture, or only shortly delayed it, could have gone unrecorded - but there are memoirs which certainly recount witnessing the occurrence. Here is an example from "I Still See Her Haunting Eyes: The Holocaust and a Hidden Child Named Aaron" by Aaron Elster:

I see a young mother breast-feeding her baby, and suddenly there is an outburst from another infant across the crowded room. The captives tell the young mother to silence the baby, but the noise continues. The elders insist that the mother must quiet the infant's outbursts, or they fear the Gestapo will discover us and we will all be murdered. The urging continues as the baby's cries become louder and louder as the onlookers become more agitated and anxious.

My eyes must be deceived me as the distraught mother places her palm over her daughter's mouth and nose and exerts pressure. The baby's legs begin to violently flair and kick until the move no more. Silence fills the room as the infant takes her final gasp for air and suddenly goes limp.

Thumb through some more memoirs, and you'll likely find more examples, but again, that doesn't give us any sense of how common it might have been, just that it happened.