Sweden’s banks have cut back on cash in part for safety reasons after a rash of violent robberies in the mid-2000s. The national psyche is marked by an infamous helicopter heist in Vastberga in 2009, when thieves landed on the roof of a G4S cash service depot and stole millions — a drama now being turned into a Netflix film. Last year, only two banks were robbed, compared with 210 in 2008.

In recent years, banks have dismantled cash machines by the hundreds. So little cash is used now that it has become expensive to track and maintain, said Leif Trogen, an official at the Swedish Bankers’ Association.

There are two proposals by the Swedish authorities to keep cash at hand. Parliament wants just the biggest banks to handle cash. The central bank is holding out for all banks to keep money flowing. Swedbank, SEB and other big Swedish financial institutions are fighting the lawmakers’ demands, saying it would place an undue burden on them to provide greater access.

“The demand for cash is decreasing at an ever faster pace,” Mr. Trogen said. “Therefore, it is fundamentally wrong to legislate to influence the demand for cash.”

The central bank has plans to roll out a pilot version next year of a new type of Riksbank money — the digital krona, or e-krona — that could replace physical cash or at least help calm the current cash conundrum. An e-krona would mean that the functions of a currency backed by the state would remain, even in an all-digital world that is fast approaching.

Christine Lagarde, managing director of the International Monetary Fund, noted last week that several central banks were “seriously considering” digital currencies. “While the case for digital currency is not universal, we should investigate it further — seriously, carefully and creatively,” she said.

Mr. Ingves, the central bank governor, said, “This is not a war on cash, but no one has argued that this evolutionary motion is going to stop.”