The library supports W5100 chip but not W5100S installed in W5100S-EVB-Pico. This library based on the official library supports the chip:

https://github.com/WIZnet-ArduinoEthernet/Ethernet

It is only necessary to add less than 10 lines of code to the official library.

Additional context

Additional requests

• Merge WIZnet fork to support more devices #185 (comment)
• Merge WIZnet fork #230