Hi, My A7Rii when set to silent mode did make a really quiet click sound. I had Audio Signals turned off so no focus acquisition beep or anything. The click I think was the AF focusing the lens. I had Setting Effects turned off on mine for battery life.
Have you got the EVF set to display Setting Effects (Gear menu page 3)? i.e. is it set to on where it shows more of a WYSIWYG image in the EVF? Reason is that that changes the lens AF behaviour. Also pre-AF or continuous AF
If they are on, the lens will stop down and move the focus etc all the time. If it is off, it operates more like a DSLR where it leaves the aperture open and when you push the shutter it stops it down for the image.
In the former situation, it will be clicking away all the time and when you press the shutter it will likely be silent because it has already done its lens adjustments.
If the latter, then you are hearing it as it stops down immediately before taking the image.
Don't know if this makes any sense but are a couple things to check.
Also, there was a faint click when you put your eye to the viewfinder, when it switches from screen to EVF. No idea why or if it still does it in Mark III.
@kallisti, possibly a test for you to do a side by side with a Sony lens?