Who was the first person condemned to death by a Christian bishop?

Priscillian, a bishop himself, was tried, condemned to death, and executed on orders of the Roman emperor Maximus in 385. The chief accusser was Ithacius, a rival bishop. Although I don't know if Ithacius desired Priscillian to be killed or if he approved of the killing, he was nevertheless censured by notable Christian figures, such as Martin of Tours, Ambrose, and pope Siricius for taking the matter to civil authorities and the result of having a bishop executed. Ithacius was subsequently removed from being a bishop.

If a bishop ever publically ordered the execution of someone it would somehow probably be in connection to the inquisition. However most inquisitors were Dominicans (or Franciscans to a lesser degree). But that said, I am sure there were probably a few bishops who may have been judges in these tribunals. When local clergy were placed in a position to judge such cases, bishops were generally not included and would be given to the lesser clergy.

The Inquisition was a group of institutions within the government system of the Catholic Church whose aim was to combat public heresy committed by baptized Christians. It started in 12th-century France to combat religious dissent, in particular the Cathars and the Waldensians. Other groups investigated later included the Spiritual Franciscans, the Hussites (followers of Jan Hus) and the Beguines. Beginning in the 1250s, inquisitors were generally chosen from members of the Dominican Order, replacing the earlier practice of using local clergy as judges. The term Medieval Inquisition covers these courts up to mid-15th century.

During the Late Middle Ages and early Renaissance, the concept and scope of the Inquisition significantly expanded in response to the Protestant Reformation and the Catholic Counter-Reformation. It expanded to other European countries, resulting in the Spanish Inquisition and Portuguese Inquisition. The Spanish and Portuguese operated inquisitorial courts throughout their empires in Africa, Asia, and the Americas (resulting in the Peruvian Inquisition and Mexican Inquisition). The Spanish and Portuguese inquisitions focused particularly on the issue of Jewish anusim and Muslim converts to Catholicism, partly because these minority groups were more numerous in Spain and Portugal than in many other parts of Europe, and partly because they were often considered suspect due to the assumption that they had secretly reverted to their previous religions. Inquisition (wikipedia)

The actual first bishop who ordered an execution of someone may be almost impossible to name.

Notwithstanding, that historical archives are not always publically available, I am going to say, for the time being, that the first person condemned to death by a bishop was St. Joan of Arc, also known as "The Maid of Orléans". The main judge in her trial was Pierre Cauchon, Bishop of Beauvais, at a time when ecclesiastical trials should have been reserved to the Dominican Order.

At the conclusion of St. Joan D'Arc's trial and condemnation, she was handed over to the secular authorities to be burned at the stack. The civil authorities were English.

The presiding judge was Peter Cauchon, Bishop of Beauvais. He was known even in his lifetime as "Peter the Pig" not only because the French word for pig, cochon, resembled his name, but also it held a definite resemblance to certain aspects of his character. He was a Frenchman, yet he was loyal to the English cause because it offered more opportunities for higher positions and acquiring more wealth. A ready and willing tool in the hands of Jeanne's enemies, Cauchon was the perfect unscrupulous and ambitious man needed for the job. This Judas Iscariot of the fifteenth century was given authority over the trial under the pretext that Compiègne, where Jeanne was captured, lay in his diocese of Beauvais. In his eagerness to make a name for himself, he was willing to take "justice" into his own hands and convict and execute an innocent girl, and for this loathsome act his name will be remembered with shame until the end of time.

On Wednesday, May 30, 1431, the day before Corpus Christi, at seven in the morning, the nineteen-year-old prisoner was told of the verdict of the court. "Will they treat me so horribly?" she sobbed, "Must my body which has never been violated be burned to ashes? I would prefer to be beheaded seven times than to be burned so." She dressed in a long white robe and a hood, and was permitted to make her Confession and receive Communion from a Dominican. That act alone was proof of the injustice of her accusers. If her judges truly believed her to be a heretic and apostate as they had condemned her, surely they would not have allowed her to receive the Holy Eucharist!

When the crowd had dispersed, and the judges had gone from the Old Marketplace, the soldiers set to disposing of the remains from the smoldering fire. At the base of the stake, mid the ashes, was found the noble heart of the Maid, intact and full of blood. In vain the English soldiers tried to burn it with oil, sulfur and charcoal, but the only remaining relic of the brave young Martyr could not be destroyed. "We have burnt a Saint!" exclaimed one of the soldiers. "We are lost." Terrified, they gathered her heart and ashes into a bag, took it to a bridge over the Seine and cast it into the river. Saint Jeanne d' Arc

Following the Retrial of Joan of Arc, Inquisitor-General Jean Bréhal drew up his final summary which describes Joan as a martyr convicted in a trial which violated Church law in pursuit of a secular vendetta. Pope Callixtus III excommunicated Cauchon posthumously in 1457 for his role in Joan's persecution and condemnation.

But then if you consider the pope as the Bishop of Rome, that is a whole different story. The first persons to be directly condemned to death by the Bishop of Rome were the brothers Carlo Carafa (Cardinal) and Giovanni Carafa, Duke of Paliano in 1561.

The first known persons directly ordered to be executed by the a pope himself seems to be the brothers Cardinal Carlo Carafa and Giovanni Carafa, Duke of Paliano, nephews of Paul IV, sentenced to strangulation in prison and beheading, respectively, by Pius IV, as his first public act (March 5, 1561). - Who was the first person ordered executed by a Roman Catholic pope?