Elizabeth Vaughan tells of turning in a manuscript in which an infant died midway through. Her publisher sternly counseled her that "In romance, you can't kill a baby." She had to rework the entire plot to accommodate the infant's survival.
While the Pern colonists' first encounter with Thread gruesomely killed several adults and at least one young girl, Dragonsdawn does honor this trope with babies. Two infants were the only survivors of the colony's Tuareg nomad camp, having been sealed inside a Thread-proof metal cabinet by their doomed parents, and a house in which a woman is giving birth was instinctively protected by hundreds of the settlement's fire-lizards.
The very premise of Harry Potter, who is The Boy Who Lived because the evil overlord wanted to kill a baby but wasn't able to. "Trying to kill a baby but not being able to" is probably the most pathetic thing a villain can do. It's hard to think of things that couldn't kill a baby, but apparently Voldy just had to get cute with the Killing Curse, instead of using the much more reliable kick to the head.
Lampshaded up the wazoo by the Samurai Cat books, in which Shiro the homicidal kittenrevels in his Baby+ Cat Immortality, gleefully rushing into meat-grinder battles in the smug confidence that the author wouldn't dare kill him. Eventually this trope was averted in Samurai Cat Goes To Hell, but only as a plot device to send his uncle to retrieve the bloodthirsty little creep.
Played straight in Breaking Dawn. While all of the Volturi converge to kill Bella and Edward's daughter Renesmee, they all instantly become captivated by her charm when they see her, quickly realize that they were wrong, and go home without a fight. Yes, that was the climax.
Completely averted in Warrior Cats. When they say Anyone Can Die, they mean it. One particular example is Snowkit, a deaf kitten, who is eaten by a hawk.
The child supervillain August Prince from Worm has this explicitly as his superpower; his presence renders people incapable of deliberately attacking or harming him.