Small companies — those with under 500 workers — employ nearly half of America’s private sector work force. Most run on thin margins and have scant savings. For restaurants, gyms and other small businesses that depend entirely on people walking in the door, sales fell to zero after stay-at-home orders went into effect.

The Paycheck Protection Program was intended as a backstop to dissuade layoffs. Under the program, small businesses could borrow up to two and a half times their average monthly payroll cost. If they used the money to retain workers and keep paying them for at least eight weeks, the loan would be forgiven in full, and they could use a portion of the cash for certain other expenses, like rent and utilities.

“We knew demand would be through the roof,” said Jim Donnelly, the chief commercial officer at Bangor Savings Bank in Maine.

The program was hastily designed after a difficult debate in Congress. Democrats had wanted the government to provide cash infusions to small businesses through tax rebates or by having the Treasury work directly with payroll processors to administer payments. They had also wanted it to cover 16 weeks of payroll, rather than eight. Republicans wanted to steer the program through private sector financial institutions. They won.

The government adapted a decades-old program run by the S.B.A. that guaranteed small business loans issued by a network of banks nationwide. That allowed the government to essentially outsource most of the program’s legwork to banks, making its adoption speedier. But the S.B.A. and the banks had never before operated a program of this scale. Last year, the S.B.A. backed around $30 billion in loans. Now, it was expected to process more than 10 times that volume, in just a few weeks.

And the administration wanted the money out the door immediately beginning on Friday, April 3. “This will be up and running tomorrow,” Mr. Mnuchin said on Thursday night. “You get the money. You’ll get it the same day.”

To bankers, that was an absurd promise. It typically took them days to prepare even the simplest loan paperwork and go over the fine print with borrowers. As Mr. Mnuchin spoke, they were still waiting for critical information and materials. For example, the S.B.A. usually required lenders to use a specific promissory note for loans it guaranteed. The agency had not yet provided that note for the new program’s loans.