Is there any country which does not have paper currency (and coins)?

In Australia, what used to be paper currency is now plastic banknotes. But we do still have coins.

Modern polymer banknotes were first developed by the Reserve Bank of Australia (RBA), Commonwealth Scientific and Industrial Research Organisation (CSIRO) and The University of Melbourne. They were first issued as currency in Australia during 1988 (coinciding with Australia's bicentennial year); by 1996, the Australian dollar was switched completely to polymer banknotes. Romania was the first country in Europe to issue a plastic note in 1999 and became the third country after Australia and New Zealand to fully convert to polymer by 2003.

https://en.wikipedia.org/wiki/Polymer_banknote

As the Isle of Man had an issue of a plastic £1 note in 1983, I'd say that beat Romania for the first in Europe. This was on Bradvek polymer (a variant of Tyvek), and came third in the world behind Haiti and Costa Rica. Australia's modern polymer notes were the first lasting polymer notes (Tyvek not being particularly successful). The quote above is from the main text of the article, but the caption below the picture of the Romanian notes has the word 'mainland' in between 'first country in' and 'Europe', which is more accurate. https://en.wikipedia.org/wiki/Polymer_banknote

As to the question itself, I can't think of a country entirely devoid of issued currency. Ecuador has given up the use of the sucre in favour of the US dollar, but mints coins in various amounts of centavos for small change. https://en.wikipedia.org/wiki/Ecuadorian_centavo_coins

Please also see https://www.funtrivia.com/askft/Question74602.html (Which is the country that has only currency notes (bills), but no coins?)

How about Sweden...

For several years now, there has been a noticeable shift away from the use of cash in Sweden. Instead, FinTech payment solutions are so widespread that the whole country is well on its way to becoming the world's first cashless society. Many shops, restaurants, cafes, and bars are cashfree - it's not uncommon to see "card only" signs next to most cash registers. Even the majority of bank branches, where you'd naturally expect to see paper notes and coins, are refusing to take cash deposits.

Cash use has declined so much that the Riksbank, Sweden's central bank, has even considered introducing its own digital currency, the "e-krona", as a complement to cash. According to the Riksbank, less than 2 percent of all transactions made in Sweden last year were made in cash. This figure looks set to drop even further to just 0.5 percent by 2020. Compare this to the global average of 75 percent and it creates a stark image of how little Swedes really use cash.

https://www.business-sweden.com/insights/articles/cashless-society/