XPS One 2720, can the processor be updated to allow Windows 11?

While the processor can be upgraded to a better performance such as i7 4th generation, it can not be upgraded to be officially support Windows 11, which requires at least 8th generation processors or higher.

So the only other option is to buy a new computer???

No, you can install Windows 11 to your system and use it until the system gives out.

Just search for how to install Windows 11 on unsupported hardware.  Although it's not officially support, those systems are still receiving Windows security updates and system updates at this current time.