This happens on Wifi and Cellular with iOS 9.1 on an iPhone 6. Clicking the update button results in the App Store somewhat ignoring the action where it seems to try updating for about 0.1 seconds (there's no way it can detect a real network failure that quickly), and then reverts back to the button as though nothing were pressed.
Repeatedly pressing the update button eventually results in the update going through, where it finishes without any issue (once it finally makes a real attempt to download it)
No other apps show any signs of having network issues and all other devices work fine on the WiFi network.
Auto-updates have been failing for a long time as well, not sure if related. "Unable to download software update" messages frequently pop-up randomly while I'm doing other things.
Best Answer
Not sure what caused the issue, but it was finally resolved by using "Reset Network Settings", which seems to do some kind of deep purge that requires rebooting the device (with a progress bar for some type of process that runs).