There are Win 3.11 drivers out for the Velocity 100/200/inofficial Voodoo3, maybe this could also work with a Banshee, because the 2D core is nearly the same?

The 3dfx 3.11 driver cames with the Velocity 100 driver CD and also works with the Voodoo 3 2000-3500, because all cards have the Avenger core.

You could try out the driver with your Banshee or looking for a Voodoo3 card.

Here are more infos and also the downloadlink for the 3.11 driver:
Windows 3.11 Drivers (Voodoo3 & #9)