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Repair the phone with XC
Alternatives on How to backup Xperias
Could you test a different router?
It's actually happening to me as well. Sometime it just stops connecting (with the WIFi on), others it simply slows the connection down. Stopping the WiFi and restarting it again in my case fixes the connection drops immediately. And yes, the other devices connected with WiFi have no issues, my Xperia M2 with.about the same apps installed (I synched them). So it seems like a kernel/drivers issue to me. Similar to the one of the Bluetooth for some (the Bluetooth never gave a problem to me, instead. And I use a BT handfree set in my car).