After updating to ICS via Sony Companion, I just encountered this same issue (however, I cannot verify the cellular network access because I don't have data) for Google Play, and in general internet access via any browser - I cannot download any of the apps I had nor update them. If I disconnect and reconnect the WiFi, it seems to work for 1 or 2 seconds, but results in the same thing thereafter. I have never encountered this problem on Gingerbread, so I don't think it's an issue with my WiFi connection, but rather ICS perhaps interfering with my WiFi connection or something along that line.
I have tried factory reset a couple of times, and even repair through PC Companion - nothing. It doesn't resolve it. I'm left with the resolve that I would need to see a Sony mobile specialist.