Martyrs die but it is not suicide. But we have to question how can that be when one refuses to do the things that will keep one alive?
Not to lessen or minimize or over-simplify the issue, but in this sense, suicide is done in despair and out of a hopelessness. It is done without freedom.
Martyrdom is done not to die but to hold fast to Christ. The problem Perpetua deals with sounds simple - just make a sacrifice to the emperor to show that you are loyal to him. But it is more - the sacrifice is not for but to the emperor; if our only king is Christ then we cannot offer our allegiance to another for any price. And unlike for the emperor, we do not make sacrifice to Christ for his benefit! What benefit can we give him? We have only our homage and worship; we benefit from this worship, not Christ; through it we show our love for him.
To not worship the emperor is to deny allegiance to him and is punishable by death. To not worship Christ bears a similar fate but at a much greater rate: "And do not be afraid of those who kill the body but cannot kill the soul; rather, be afraid of the one who can destroy both soul and body in Gehenna." (Matthew 10:28)
So martyrdom shows "fear of the Lord", not fear of death. Perpetua and Felicity show that it is worse to abandon Christ then it is to abandon life, those we love, and who love us. But it is the same end that we all share whether by martyrdom or natural causes. The difference is where you end up. Death has no more power over us - do we believe that?
One day while we were eating breakfast we were suddenly hurried off for a hearing. We arrived at the forum, and straight away the story went about the neighborhood near the forum and a huge crowd gathered. We walked up to the prisoner's dock. All the others when questioned admitted their guilt. Then, when it came my turn, my father appeared with my son, dragged me from the step, and said: 'Perform the sacrifice--have pity on your baby!'
Hilarianus the governor, who had received his judicial powers as the successor of the late proconsul Minucius Timinianus, said to me: 'Have pity on your father's grey head; have pity on your infant son. Offer the sacrifice for the welfare of the emperors.'
'I will not', I retorted.
'Are you a Christian?' said Hilarianus.
And I said: 'Yes, I am.'
When my father persisted in trying to dissuade me, Hilarianus ordered him to be thrown to the ground and beaten with a rod. I felt sorry for father, just as if I myself had been beaten. I felt sorry for his unhappiness in his old age.
Then Hilarianus passed sentence on all of us: we were condemned to the beasts, and we returned to prison in high spirits. But my baby had got used to being nursed at the breast and to staying with me in prison. So I sent the deacon Pomponius straight away to my father to ask for the baby. But father refused to give him over. But as God willed, the baby had no further desire for the breast, nor did I suffer any inflammation; and so I was relieved of any anxiety for my child and of any discomfort in my breasts....
Some days later, an adjutant named Pudens, who was in charge of the prison, began to show us great honor, realizing that we possessed some great power within us. And he began to allow many visitors to see us for our mutual comfort.
-- From the Acts of Perpetua and Felicitas