Tertullian writes in his Apologeticum (40.2):

If the Tiber reaches the walls, if the Nile does not rise to the fields, if the sky does not move or the earth does, if there is famine, if there is plague, they cry at once, "The Christians to the lions!"

The Passion of the Holy Martyrs Perpetua and Felicity is an account of a young noblewoman and mother and a slave woman who were martyred by damnatio ad bestias in early third-century Carthage. Lions are not specifically mentioned, but other animals are. It is a short account. Here is the text.

We know lions and other big cats were used to execute insurrectionists and criminals in Rome (Zliten Mosaic), and we know Christians were executed by beasts. The extent of the practice and the specific accounts of what animals were unleashed is what is debated.

Christians were executed in a variety of ways. Eusebius records in his De Martyribus Palestinae that Urbanus, governor of Palenstine, had martyrs burned, drowned, thrown to beasts, and some youths forced into gladiatorial combat, along with other forms of torture and imprisonment.

I actually just answered a similar question a few days ago, so here is the link to the full answer. However, I copy and pasted most of it down below and made edits to specifically fit your answer.

To answer your question, we have to ask "Were Christians persecuted the way that the Christian Churches explain it today?" This is a valid question since Christian tradition teaches that Christian's came under heavy persecution throughout their early existence. Being martyred and being thrown into gladiatorial battles is often envisioned and cited whenever I ask my students (I'm a teaching assistant at a university) what they thought the early years were like. However, this isn't the most reasonable assessment nor is this belief held by most scholars.

A truly excellent book on this topic was by historian Candida Moss, whom wrote The Myth of Persecution: How Early Christians Invented a Story of Martyrdom. Here, Moss (a professor of New Testament Studies at Notre Dam) argues that there was never a wide-spread persecution of Christians as most people believe. Of the over 250 years of Early Christian history, Moss presents evidence that at most there were only 12 years (at most) where people were harshly persecuted for their faith by orders of the Roman Empire (none of them coming in the first century, during the early conversion times). For it's first few decades, Christians probably went relatively unnoticed. Small churches were known to have sprung up throughout the eastern Mediterranean and even at least one in Rome by 70 CE (this is about when and where Mark wrote his gospel), but it is unlikely Christians faced any real threat yet since they weren't converting huge percentages of people in their populations.

As Christianity became more popular, most likely by the end of the first century, it is likely that this is when they started facing harsher criticism. Most were simply scorned or disliked, similar to hows Jews were treated once they left the Levant. For most early Christians, "the vast majority of Christians never stood before a Roman judge, paid a fine, or experienced torture," [Moss, page 129] however, some did. Moss (and other historians) have pointed out that persecution did take place in several locations around the Roman Empire, but again, this is more the exception to the rule. The reason for this is simple, "For almost all of the first century, it’s unclear that Roman emperors even knew that Christians existed," [Moss, page 139] meaning that you can't persecute and hate something that you didn't even know existed.

I will add that there were some specific areas and period of persecution where select numbers of Christians were tortured or even killed, but this didn't happen until the second century, and was rare and didn't last for very long. For the most part, Christianity was a small sect that didn't expand considerably until Constantine, and for the most part, they didn't really pose as a serious threat to Rome.

Now for those Christians whom were unlucky enough to face serious persecution, there were several ways that they met those ends. One of the ways would be to be fed to large beats, but they could also face torture, crucifixion, or any of the dozen or so other ways that the Romans typically killed their oppositions.

TL:DR: While it is likely that some Christians were killed by lions or other similar beasts, it rarely happened over the first two and a half centuries of Christianity's existence.