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Sounds strange. Have you tried to just restart the phone?
I suggest that you clear data for Chrome under settings > apps > all > chrome > clear data. Restart the phone and try again.
Strange! Try to remove the battery from the phone for a while and then try again. You can also start your phone in safe mode to see if the problem may be related to any 3rd party apps installed in your phone:
http://talk.sonymobile.com/t5/FAQ/How-to-boot-your-phone-into-safe-mode/m-p/348008#U348008
Do you experience the same problem if you download another browser from Google Play?
I've got te same problem With my xperia L. I uninstalled chrome and installed dolphin but still no internet connection. no facebook, nothing at all.
just solved te problem. The update deleted all my internet settings 😞