Playing continuous signal to keep it from clicking or sleeping is a great idea, but eventually the squirrel will get his nut and you'll hear the click again.
It's not sleep, it's just the DAC takes more time to sync to the digital input signal than it should. This is normal for older DACs. It's the syncing and unsyncing of the digital input signal. See if it also snaps when you unplug the digital input cable. Some music player software allows setting a delay to starting the music playback to allow the DAC to sync first. Usually the problem is only missing the first notes, not a transient snap. That sounds like something broken.
If it makes a snap sound AFTER the music stops while playing nothing, then you may have a DC offset problem with your DAC. You can measure the output DC offset just to make sure it's not physically broken. Measure output of DAC with multimeter set to VDC. Should be <0.050VDC at all times: while playing music, while playing silent and also during sleep should be no DC on the output. It might be snapping to a DC offset during sleep.
If you have a very high sensitivity speakers (>95dB) then every flaw in the signal will be very audible and you need to upgrade to quieter electronics, or lower voltage gain amplifier, etc.