No.

At least three South Dakota governors attended and graduated from private high schools before delving into politics.
Gov. Larry Rhoden graduated from Sunshine Bible Academy in Miller and governors Archie Gubbrud (served 1961-65) and Sigurd Anderson (1951-55) graduated from Augustana Academy, a Lutheran high school in Canton that closed in 1971.
Rhoden attended the private school in Miller, which opened in the early 1950s. The school is still in operation and averages 15 students per grade level.
Augustana Academy attracted students from across the country and had a graduating class of 63 in 1959. Other notable alumni include TV personality Mary Hart (graduated 1968) and Ole Rolvaag (early 1900s), author of "Giants in the Earth.”
Former Gov. Bill Janklow did not graduate high school. He dropped out of Flandreau High before his junior year and joined the U.S. Marine Corps, later attending the University of South Dakota.
This fact brief responds to conversations such as this one.
Sources
Sioux Falls Argus Leader, Whatever Happened To: Augustana Academy in Canton
Gov. Larry Rhoden, Address to Joint Legislative Session
Sunshine Bible Academy, About the school
Sunshine Bible Academy, FAQ
Rapid City Journal, Timeline of Bill Janklow's life and career
Trail of Governors, Sigurd Anderson and Archie Gubbrud
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