According to the user guide, the camera has an internal rechargeable battery that maintains the date and time settings (plus other user defined settings) when the camera is switched off and regardless of whether the main battery is charged or not.
Here is a link to the ifixit [[[Topic:Sony Cyber-shot DSC-WX350]] guides for your camera. Unfortunately they do not show where the battery is located. Perhaps @oldturkey03 may be able to supply the camera's service manual which will help you to locate and replace the battery
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=== Update (10/17/2017) ===
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Hi @didiere ,
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To give you some idea here is an answer for the same problem with a Sony CyberShot DSC-WX230 showing where the battery is located and what type it is.
Hi @didiere ,
According to the user guide, the camera has an internal rechargeable battery that maintains the date and time settings (plus other user defined settings) when the camera is switched off and regardless of whether the main battery is charged or not.
Here is a link to the ifixit [[[Topic:Sony Cyber-shot DSC-WX350]] guides for your camera. Unfortunately they do not show where the battery is located. Perhaps @oldturkey03 may be able to supply the camera's service manual which will help you to locate and replace the battery