Yes, indeed it does.

Nebuchaadnezzar sacked Jerusalem twice, first in 597/598, and a second time in 587/586 BCE, according to Babylonian records of the time (see Wiseman,D. J., 1956 Chronicles of Chaldean Kings in the British Museum London: Trustees of the British Museum).

After he conquered them the first time, he set up a puppet king, but this did not work out and the Judahites revolted, and so he conquered them a second time, and this time he deported a large number of the inhabitants of Judea back to Babylon (see Vanderhooft, "Babylonian Strategies of Imperial Control in the West" in Lipschits, O., & Blenkinsopp, J. (2003) Judah and the Judeans in the neo-Babylonian period. Winona Lake: Eisenbrauns. We know this from several sources: Babylonian records show the presence of large numbers of Israelites in Mesopotamia (such as theophoric names, or names which also contain the name of a deity, in this case Yahweh, indicating that these people worshipped the Judahite God). In addition, there were enclaves of people who had been deported back to Babylon, and the one where the Judahites ended up was called al-Yahudu, which could be translated into modern common parlance as "New Judahtown" (see K. Abraham (2010) "The Reconstruction of Jewish Communities in the Persian Empire: The Āl-Yahūdu Clay Tablets" in Agai Segev and Asaf Schor (eds.), Light and Shadows - The Catalog -The Story of Iran and the Jews (Beit Hatfutsot), Tel Aviv 2011).

Even the end of the Babylonian Captivity as recorded in the Bible has a historical basis. The Cyrus Cylinder records King Cyrus of Persia's efforts to restore the peoples deported by the Babylonians.

So yes, there is extensive extrabiblical evidence supporting the Biblical account of the "Babylonian Captivity".

The Bible is often discounted as a historical source, when it should not be. It should be treated the same way as any other ancient text, written from the point of view of a specific people group and written for a specific purpose (in the Bible's case, the purpose is theological). It's not a history textbook, but no ancient text is.