This article is from the Calendars FAQ, by Claus Tondering firstname.lastname@example.org with numerous contributions by others.
If you define a "week" as a 7-day period, obviously the answer is
no. But if you define a "week" as a named interval that is greater
than a day and smaller than a month, the answer is yes.
The ancient Egyptians used a 10-day "week", as did the French
Revolutionary calendar (see section 7.1).
The Maya calendar uses a 13 and a 20-day "week" (see section 8.2).
The Soviet Union has used both a 5-day and a 6-day week. In 1929-30
the USSR gradually introduced a 5-day week. Every worker had one day
off every week, but there was no fixed day of rest. On 1 September
1931 this was replaced by a 6-day week with a fixed day of rest,
falling on the 6th, 12th, 18th, 24th, and 30th day of each month (1
March was used instead of the 30th day of February, and the last day
of months with 31 days was considered an extra working day outside
the normal 6-day week cycle). A return to the normal 7-day week was
decreed on 26 June 1940.