Have you tried to clean the lens, both front and back...as in taking it off the camera...using can air and blow it clean. also do the glass element on the front of the camera...many years ago, this happened on my Sony BVW-300a betacam...and it was just dust. BUT...it could also be a "dead pixel"...if you put the camera on 6db gain and if it gets "brighter"....not good....try a good cleaning first....
Definitely sounds like a "lit pixel", but you can likely fix it yourself.
Try multiple Auto Black Balances (ABB), or else search you menu for Auto Pixel Correction (APC). I'm not familiar with that particular camera, but some Sony's have it on the Maintenance page.
Both or these methods essentially identify the lit pixel and then just "turn it off". There is a buffer that holds the addresses of the pixels you've turned of. Eventually this buffer might fill up. That's the time you have a real issue, as sometimes you have to actually replace the sensor, which is not a practical consideration in an inexpensive camera.
If it is dirt, you might be able to tell by stopping the iris down, this increases the depth of field and reveals the dirt, in, on or behind the IR filter of sensor cover glass. There are kits to actually inspect the the sensor and clean it (usually marketed to DSLR users) but you have to be able to remove the lens to gain access for cleaning. If you can't it'll likely be a warranty issue, as it's supposed to me a "sealed" area where dirt cannot get to. If you can remove the lens, do so and shine a flashlight in there, and look at the picture on a 17" or larger monitor, you'll see the dirt if that's the culprit. Gaining up will reveal lit pixels.