Can I have more than one bank connected to Apple Cash?

No. You can only have one at a time, but you can change it at any time.

See the section titled "How to update your bank account information" in the following support document.

click here ➜ Transfer money in Apple Cash to your bank account or debit card - Apple Support

Currently, that’s not an option. You could use a debit card from one account and a checking account that is a different account. But that limits you as to which account receives instant transfers and which account receives ACH transfers.