The short answer is: No, the maximum practical color depth for images on websites is 24-bit color. Only a few image formats support a higher color depth (TIFF is a remarkable example), and there are none of these that are supported by all common web browsers.
Wikipedia offers a table with web browsers and the image formats they support. You can see that Chrome and Firefox, which account for about two-thirds of the browsing, don't support TIFF, although Safari and Edge do. Unfortunately, even if you provide a reliable mechanism for delivering TIFF files to browsers that support them and JPEG to the rest, your TIFF files will only be seen by a third of the visitors.
Aside from browser support, there is another limitation: Many people don't have display systems that can display more than 16.7 million colors. This is not just a function of the monitor, since the graphics card must also support a higher color depth. These are both more expensive and specialized items that most don't have on their desks at home.
Even if browser support for deeper colors increases, the hardware standards have to be changed massively before they become generally accessible.