Character set that is not a superset of ASCII

Yes. JIS X 0208 is not a superset of ASCII. Some versions of this standard include most of the ASCII characters, but not all of them.

A related fact is that a file encoded with UTF-16 or UTF-32 is not byte-equivalent to an ASCII file of the same characters, but since those are not character sets, and since Unicode is certainly a superset of ASCII, they do not qualify as answers to your question.

There were some ISO 646 derivatives that were basically ASCII, but replaced various punctuation marks with various accented characters. There were also Greek, Cyrillic, Arabic and Hebrew sets that laid those characters over the Latin characters; see http://en.wikipedia.org/wiki/ISO_646 for details.