The TV's WiFi radio probably has a driver issue or the connection settings are being reset somehow. Since your internet works fine everywhere else, it's not a network problem - something on the Sony itself is flaking out. Start by going into the TV's network settings and forgetting the WiFi network entirely, then reconnect from scratch. While you're in there, check if there's a firmware update available - sometimes Sony pushes out fixes for WiFi stability issues.
If that doesn't stick, try moving the router closer to the TV temporarily to rule out a weak signal problem, even though it seems unlikely. Also check if the TV is placed near a microwave, cordless phone, or other 2.4GHz devices that might be causing interference. A lot of times just repositioning stuff clears it up. If none of that works, you might need to do a factory reset on the TV itself - it'll wipe everything but can fix stubborn connection bugs that partial restarts miss. Just make sure you know your WiFi password before you do it since you'll need to set it up again.