WASHINGTON — Millions of taxpayers who waited until Tuesday to file their 2017 tax returns and make payments through the Internal Revenue Service’s website were thwarted by a systemwide computer failure that advised last-minute filers to “come back on Dec. 31, 9999.”

The website malfunction, which began in the early hours of Tuesday morning and was not resolved until early evening, crippled a crucial part of the agency’s website that allows taxpayers to file returns electronically and make their tax payments directly through their bank accounts. The technology failure essentially brought the nation’s tax machinery to a halt on a day when millions of Americans were expected to file their tax returns, undermining the Trump administration’s plan to use Tax Day to promote its recent $1.5 trillion tax overhaul.

The I.R.S. said in a statement that taxpayers would be given an additional day to file their returns.

On Tuesday, taxpayers could still file the old-fashioned way: by sending their returns and payments through the mail. But for most of the day, those seeking to make a payment online ahead of the midnight deadline were greeted with the message: “This service is currently unavailable.” The website said it was undergoing a “planned outage” beginning on Tax Day that would last until Dec. 31, 9999, but said tax payments were still due on April 17, despite the system’s inability to process them.