A police source also shared a video footage, in which two officers are seen leading a handcuffed man to the police station.
Authorities have not released the name of the attacker. Several local media reported his age as 25 and his nationality as Moroccan.
The deceased was identified as Diego Valencia, an employee of the Nuestra Señora de la Palma church.
The attackers also attacked the nearby San Isidro church, where priest Antonio Rodríguez was among the wounded, in critical condition, Alhefirat city spokesman said.
Authorities did not say how many people were injured in the attack. The newspaper El Mundo reported that the number of injured was 4.
Police say the suspected assailant first entered the San Isidro church and attacked 74-year-old Rodriguez; He then went to the church of Nuestra Señora de la Palma, destroyed its contents and attacked Valencia.
At one point, the assailant chased after Valencia, who managed to get out of the church and hit him again, the police statement said.