Is Vertical Speed based on True Airspeed or Groundspeed?
Vertical speed, as measured by the Vertical Speed Indicator (VSI) should be measured entirely by the changing air-pressure at the static port, and should be completely unrelated to airspeed and ground speed.

The static port is a small hole, (about the size of a large needle or tiny nail) in the side of the airplane, with a pressure-sensor inside.  As that sensor measures a decreasing pressure, it indicates that the airplane is climbing.

An increasing pressure indicates the airplane is descending.

Neither measurement should depend on the plane's air or ground speed.  The sensor should still work even if mounted on a hot air balloon (or other stationary craft).

Source: http://www.boldmethod.com/learn-to-fly/aircraft-systems/how-does-a-vsi-work/