Is a pulse of white noise still properly described as stationary? I understand that a signal that is white noise is stationary (or more properly that the process generating it is stationary).

What if the white noise is delivered as a single pulse or a series of pulses with silence between them? Is that still stationary, or does the pulsatility suddenly convert this into a non-stationary system?

If it is no longer stationary, is there a term of art conveying the information that the signal within the pulse is stationary (e.g., a pure tone or white noise) rather than  a non-stationary natural or musical sound?
The pulsed process is not stationary. You can use the term "quasi-stationary" to describe a process that looks stationary in short time scales (e.g. during silence, or in the middle of a pulse).