Into what was Jeremiah lowered with ropes, where he sank into the mud?

Answer: A cistern

Officials dropped him into the cistern of Malchiah, which had no water, only mud (KJV: "dungeon… mire").

“So they took Jeremiah and dropped him into the cistern of Malchiah, the king’s son, which was in the courtyard of the guard. They lowered Jeremiah with ropes into the cistern, which had no water but only mud, and Jeremiah sank down into the mud.”— Jeremiah 38:6 (BSB)