A non-fungible token (NFT) market is a new trading invention based on the blockchain technology, which parallels the cryptocurrency market. In the present work, we study capitalization, floor price, the number of transactions, the inter-transaction times, and the transaction volume value of a few selected popular token collections. The results show that the fluctuations of all these quantities are characterized by heavy-tailed probability distribution functions, in most cases well described by the stretched exponentials, with a trace of power-law scaling at times, long-range memory, persistence, and in several cases even the fractal organization of fluctuations, mostly restricted to the larger fluctuations, however. We conclude that the NFT market—even though young and governed by somewhat different mechanisms of trading—shares several statistical properties with the regular financial markets. However, some differences are visible in the specific quantitative indicators.