Thanks James.
I will give your screensaver suggestion a try and see how that works. Sleep is working fine so far though, since I rolled back to the April driver.
I haven't seen a June driver. There is a driver in the Surface Pro 3 June driver download package ZIP file, but the Marvell driver in there is a May one - the same which is installed by Windows update, I believe. If there's a June Marvell driver, I'll seek that out.
It's not my router, I'm pretty sure of that. The problems happen on both my home WiFi and my corporate WiFi (over which I have no control anyway).
There's no sleep/power management options under the WiFi adapter in device manager.