Jesus Christ, on earth, was the incarnate Word: the Second Person of the Trinity, the Son. His earthly body and the soul which animated it were both at once human and divine.
To say “God died” actually denies the human soul and its continued existence after the death of the body. Even though Jesus’ body died, his soul did not — or what hope is there for the rest of us?
His body died, but his soul — with its divine nature as well as human — continued its existence. The Word did not die. Indeed, not only did the Word not die, but he descended to hell. The gates of hell in Mt 16:18 are important because they could not contain him.
The Catholic Catechism puts meat on the bones of this answer:
625 Christ's stay in the tomb constitutes the real link between his passibleA state before Easter and his glorious and risen state today. The same person of the "Living One" can say, "I died, and behold I am alive for evermore":465
God [the Son] did not impede death from separating his soul from his body according to the necessary order of nature, but has reunited them to one another in the Resurrection, so that he himself might be, in his person, the meeting point for death and life, by arresting in himself the decomposition of nature produced by death and so becoming the source of reunion for the separated parts.466
626 Since the "Author of life" who was killed467 is the same "living one [who has] risen",468 The divine person of the Son of God necessarily continued to possess his human soul and body, separated from each other by death:
By the fact that at Christ's death his soul was separated from his flesh, his one person is not itself divided into two persons; for the human body and soul of Christ have existed in the same way from the beginning of his earthly existence, in the divine person of the Word; and in death, although separated from each other, both remained with one and the same person of the Word.469
627 Christ's death was a real death in that it put an end to his earthly human existence. But because of the union his body retained with the person of the Son, his was not a mortal corpse like others, for "divine power preserved Christ's body from corruption."470 Both of these statements can be said of Christ: "He was cut off out of the land of the living",471 and "My flesh will dwell in hope. For you will not abandon my soul to Hades, nor let your Holy One see corruption."472 Jesus' Resurrection "on the third day" was the proof of this, for bodily decay was held [by the people of the time] to begin on the fourth day after death.473
A passible: able to suffer [from Latin passio, suffering]
465 Rev 1:18.
466 St. Gregory of Nyssa, Orat. catech. 16: PG 45, 52D.
467 Acts 3:15.
468 Lk 24:5-6.
469 St. John Damascene, De fide orth. 3, 27: PG 94, 1097.
470 St. Thomas Aquinas, Summa Theol. III, 51, 3.
471 Is 53:8.
472 Acts 2:26-27; cf. Ps 16:9-10.
473 Cf. I Cor 15:4; Lk 24:46; Mt 12:40; Jn 2:1; Hos 6:2; cf. Jn 11:39.