Early Rhode Island Bank Notes
1808 Ten Dollar Banknote from Farmers Exchange Bank Gloucester
1857 One Dollar Banknote from the Tiverton Bank

In 1809, the Farmers Exchange Bank of Gloucester became the first bank in the United States to fail.

The Tiverton Bank also failed, however I don't have the date of the closing.
(Possibly between 1865-68 - See Below)

In 1865, Congress imposed a prohibitive 10% tax on State Banknotes which effectively stopped State Banks from issuing banknotes.

It was generally believed that this would make the public better off by doing away with inferior brands of currency while also helping to finance the Civil War by enhancing bond sales. However, the actual purpose of the 10% tax was not to improve the quality of the currency, nor to enhance bond sales. National currency and banknotes were suffering from high inflationary pressures due to the expense of the Civil War. The actual purpose of this tax was to offset the effects of inflation on the National Banknotes.

The enactment of this tax not only drove the State Banks out of the business of producing Banknotes, but also forced many of the State Banks to become members of the Federal Bank system.  In 1863 there were 1466 State Banks in existence, however by 1868 there were only 247 remaining.