The Book of Isaiah is composed of three sections.
Chapters 1-39 - written by Isaiah (late 8th century BCE), though passages about the downfall of Babylon actually relate to Assyrian dominance over Babylon during the Neo-Assyrian period which began in the 8th century BCE during the reign of Tiglath-Pileser III. These are frequently retroactively ascribed by believers to the Persian defeat of Babylon (where the fact that Babylon is still inhabited is either ignored or glossed over with various trite excuses).
Chapters 40-55 - written near the end of the Babylonian exile or shortly after the initial return in 538 BCE. This section includes reference to Cyrus and the Persian defeat of Babylon.
Chapters 56-66 - a post-exilic work.