Because conditions varied wildly over time and across the different camps (political prisoner camps, forced labour camps, ghettos), I am going to concentrate on Auschwitz as there is rather a lot of material available about this harrowing question.

Warning: this is not pleasant reading

Jewish women

Before the autumn of 1944, when systematic gassing of Jewish inmates was halted, all Jewish babies were killed upon birth, generally together with the mothers who were guilty of the “crime” of arriving or falling pregnant in Auschwitz. If the pregnancy was discovered before the birth, the women were killed too. This led to the drama of improvised abortions and concealed births followed by infanticide, either by the hands of the mothers or by the physicians, nurses or midwives among the inmates that were assisting them in their labour. The most famous of these doctors was Gisella Perl, a Jewish-Romanian gynecologist who wrote I was a doctor in Auschwitz in which she describes how she performed many abortions to save the mothers' lives.

After October 1944, Jewish babies were not automatically killed, but this didn't increase their chances of survival significantly, as no accommodation was made for the welfare of mother and child, and the women were expected to continue with the excruciatingly hard work and subsist on literal starvation diets. There are only eight recorded births of Jewish babies in Auschwitz. There is no record of any surviving.

The family camps

There were two “family camps” at Auschwitz where certain groups were allowed to live on as best they could on starvation rations and racked by diseases caused by overcrowding and unsanitary conditions. The inmates in these "family camps" were not subjected to wholesale gassing of children, the sick and the elderly upon arrival as the regular transports were and families were allowed to stay together.

The “Gypsy” camp was established before it was finally decided that these people were all to be exterminated too. There were sporadic gassings, though. It housed Sinti and Roma families from February 1943 to August 1944. Occasionally, groups of inmates were sent to other camps for forced labour. In August 1944 almost all the remaining inmates were killed. More than 370 children were born in this camp, though it is unclear whether any survived.

The Theresienstadt family camp was in operation from September 1943 to May 1944 and was part of the whole Theresienstadt propaganda effort to “prove” to the outside world that Jews were not being killed after deportation. It housed families deported from the Theresienstadt ghetto in Bohemia who were forced to write letters about how good they had it at Auschwitz and that the families were staying together. Pregnant women were allowed to give birth. However, after six months the camp was liquidated to make room for new transports from Theresienstadt, and these inmates in turn were all killed in July 1944.

Non-Jewish and non-”Gypsy” women

Most of these women were Polish and Soviet “political” prisoners, though there were some German inmates (Jehova's witnesses, women convicted of crimes, prostitutes, etc) as well as a smattering of “political” prisoners from other countries. Policies were more erratic here. At first, these women were killed upon arrival if they were found to be pregnant. If they fell pregnant after entering Auschwitz, they generally resorted to secret abortions much in the same way as did the Jewish women. Starting from 1943, women were allowed to give birth, but many babies were subsequently killed, sometimes immediately, sometimes later, depending, it seems, on the whims of the SS. Generally, the women were forced to kill their own babies, or this was done by the medical staff who were inmates themselves. However, some blond and blue-eyed babies were taken away to the Potulice concentration camp or similar places that acted as transit camps for Polish children who were deemed to look “Aryan” enough to be subsequently adopted by German couples. In September of 1943, the first baby was officially registered as a camp inmate and received the distinctive Auschwitz tattoo with its inmate number. At liberation there were 156 children of less than three years still alive in Auschwitz, but it is not known how many (if any) of these were actually born there (a number of children were sent to Auschwitz in the wake of the Warsaw uprising of autumn 1944). The living conditions were such that a baby had very little chance of survival.

Sources:

Langbein, Hermann. People in Auschwitz. University of North Carolina Press, 2004.

Sinti and Roma (Gypsies) in Auschwitz

Bársony, János, and Ágnes Daróczi, eds. Pharrajimos: the fate of the Roma during the Holocaust. IDEA, 2008.

Gutman, Yisrael, and Michael Berenbaum, eds. Anatomy of the Auschwitz death camp. Indiana University Press, 1998.

It partly depends on what you mean by "at a camp." There were lots of different kinds of camps. Certainly there were babies born in ghettos, such as those in Warsaw or Lodz. Equally certainly, there weren't any babies born in extermination camps like Chelmno or Treblinka - their inmate populations were very small, only a few dozen, maybe a hundred or so sonderkommandos that were killed and replaced regularly.

What you're probably thinking of are the slave-labour camps - the ones that provided many of the archtypical images of the Holocaust. Prisoners starved to the point of looking like skeletons, shaved heads, striped pajamas, that sort of thing. Here's where it get's a little bit difficult to say. I've heard anecdotes of female prisoners giving birth, but it's always secondhand, and personally do not know of any reliable reports of this. It is important to understand that the prisoners in these camps were all on starvation rations - most of the women in the camps were so malnourished that they weren't having their periods. It wasn't uncommon for girls who were liberated at age 16 or 17 to not have had a single menstrual cycle by that point in their lives, having been consistently malnourished for years. Plus, pregnant women or women with young children were as a rule not sent to work - they were usually sent to the gas chambers.

There's a ton of great books on the subject. If you want memoirs of life in the camps, Night by Elie Wiesel is of course the most famous one, though I would actually recommend Survival in Auschwitz by Primo Levi over Night. Glancing at my bookshelf, a couple of works by historians that are catching my eye are Nazi Germany and the Jews by Saul Friedlander or Bloodlands by Tim Snyder.

Gisela Perl wrote a book on her time as a doctor in Auschwitz in which she admits to performing hundreds, if not thousands, of abortions. Her story seems to imply that pregnancy was not unusual and that many in the camps knew that they would be at risk by continuing the pregnancy. Starvation rations would make the pregnancy difficult at best and dangerous at worst. Furthermore, a pregnant inmate would be classed as unfit for work and would have been killed. What Perl stresses is that many of the pregnancies were the result of consensual relationships, with some in the camp effectively marrying in the face of danger.

Gisela Perl, I was an Auschwitz Doctor

Such stories are rare because pregnant women or small children in the death camps were often sent directly to the gas chambers, or even subject to grotesque "medical" experimentation. The only stories I have heard of the infants surviving all involved women who gave birth within days of liberation.

Anka Bergman was one such mother. She gave birth in Auschwitz. Her story is easy to find in a number of publications, like this Guardian piece.

This is another story about a survivor named Miriam Rosenthal who avoided a culling of pregnant women at Auschwitz and became one of seven women to give birth at Kaufering, a subcamp of Dachau. When the article was published in 2012, all seven children were still alive.