Interestingly, the University of Illinois at Urbana-Champaign actually has a sinking library. Or rather, a sunken library. The story is so often recited on campus that a school a cappella group has their own song about it.
See, the UIUC started its life as an agricultural school. They taught liberal arts as well, like any good school, but agriculture research was a big part of their mission. (What else would you expect from a state university located in the middle of the Corn Belt?) In 1876 they established the Morrow Plots to test how well corn grows in varying soil conditions. The Morrow Plots are still there today, although now they're barely larger than the ground area of two dorm rooms, marked with a plaque advertising them as the oldest experimental agriculture plot in America. It's a small source of pride for the University, which is why it's still there.
Fast-forward to the middle of the twentieth century, when the University decided to build a new library for casual undergraduate use. (Besides the main library building, UIUC has literally dozens of small libraries in various buildings for specific disciplines, and all together they compose the third-largest library in the United States today). They area marked for this new library was right next to the Morrow Plots, specifically on the west side. Members of the agriculture departments insisted that a tall building couldn't be put there because it would block the setting sun's light from reaching the Plots.
So they sunk the library instead. The undergraduate library today consists of two below-ground floors connected to the main library by a tunnel, and connected to the outdoors by a low entryway just large enough to comfortably contain the stairs leading down.