I'm not an expert for the RP2040 board but if the corresponding core doesn't support a flash file system (p.e. the SPIFFS on the ESP32) the library cannot provide such a functionality. The Arduino-Pico core supports the LittleFS for such purposes but don't expect any example of a general purpose library as the Ethernet library to support stuff available on a single platform only.
BTW, how do you want to use the Ethernet library on that board? Did you attach a WizNet board to it?