No. Once Light dies in the Death Note manga, that marks the end of the core story. There may be spin-offs or related works, but the main narrative doesn't carry on from that point.
It focuses on the story of L's early days and how he solves cases. In this novel, we see L's unique detective methods and his battle of wits against various criminals.
One possible fanfiction could be where Light is in the hospital due to an unexpected illness. Maybe he uses this time to plot his next moves in the Death Note world without arousing too much suspicion. He could be pretending to be weaker than he actually is while observing the people around him, looking for potential Kira supporters or enemies among the hospital staff and patients.
Another main feature is that they can be a great addition for fans who want more of the 'Death Note' world. They are written in a way that complements the existing media, whether it's the anime or the manga. They might introduce new characters or scenarios that still fit within the overall 'Death Note' theme, like the new mysteries and investigations in some of the novels.
Light dies when his name is written in the Death Note by Ryuk. Ryuk, who was the shinigami that gave Light the Death Note, decided to end it as per the rules.