Dead, no power on

"@geomorzak there is no specific part that may have failed. You do want to make sure that your battery is okay and that the battery makes proper contact. Make sure that the battery cover closes properly and that the battery contacts are not bend, broken or corroded.

After that you are bound to have to take the camera apart. You can start with this guide Panasonic Lumix DMC-FZ35 LCD Replacement to at least get you into the camera. Of course you can also use these instructions DMC-FZ35-Disassembly.pdf

Now check your circuit boards for any damaged and/or missing components. Pay particular attention to the main PCB around IC1001 which is your power supply IC. Post some images of your circuit boards once you have it apart. We can always try and help you with further troubleshooting. Use this guide Adding images to an existing question for that."