As per the link posted by Brian Z ("One Time Fits All" by Ian R. Bartky), the Pitcairn Islands could have been this at some point between 1790 and 1808; during this period they "lost their daily reckoning" before being given the date by the crew of the Topaz, but there's no indication they were observing any form of daily calendar by then. The Topaz gave them the date on the western (Asian) side of the date line instead of the eastern side, so the Pitcairn Islanders' count was one day ahead until 1814, but as that's date line shenanigans it doesn't count.
However, there is a real and verifiable example of a community observing the seven-day week out of sync with everyone else: the Taiping Heavenly Kingdom. Founded in 1851 as a result of the Taiping Rebellion, at their peak they held several provinces worth of southern China, including Nanjing which became their capital.
In 1852 they instituted a solar calendar, in which months alternated between 31 days and 30 days in length. They also adopted the seven-day week and observed Sabbaths -- but they were one day ahead of everyone else. Quoting from "The Taiping Revolutionary Movement" by Jian Youwen:
This revolutionary calendar, in many ways the most significant cultural achievement of Taiping Tienkuo, nevertheless suffered from two initial miscalculations. The first, still unaccounted for, put the Taiping calendar one day ahead of the Chinese lunar and Western solar calendars, e.g. the Taiping Sunday was Saturday in the Western calendar. This error was never corrected. The second miscalculation, or rather oversight, related to the difference between the actual solar year of 365 1/4 days and the Taiping calendar year of 366 days. No provision was made for the surplus 3/4 day each year with the result that there was a cumulative retardation in the beginning of solar terms amounting to 30 days in the course of 40 years. To allow for periodic correction, Hung Jen-kan years later as premier in Nanking decreed a special short year in every 40 whose months would have only 28 days each.
The calendar never reached year 40; the kingdom fell in 1864, though isolated rebel groups continued to exist for several years afterwards. (Incidentally, the calendar also used the Chinese sexagenary cycle to label days, but that was also one day off.)
More detailed sources are available in Chinese; I found this linked from the Chinese Wikipedia page on the calendar which talks about the discrepancy and its possible causes at length (and suggests that there were rebel armies using the calendar as late as 1869) but I can't vouch for machine translation accuracy.