Have you contacted the USGenWeb site for Wisconsin > counties > Bayfield to narrow your search of the censuses? The families had to be there, but I would guess that the enumerators or transcribers mangled the spelling. The Bayfield contact for GenWeb should be able to help.

An obituary is online for Thomas BEEKSMA (7 Apr. 1909, Canada - 27 June 1999). He was son of Onne & Sybil BEEKSMA He is also listed on Social Security Death Index and FindaGrave. The Canadian census of 1906 shows BEEKSMA / BECKSMA / BAXMA / BOXMA, etc. of Selkirk & Winnipeg, Manitoba. I also see on the US Army list from Clover, Lucas & Opie BEEKSMA. On the county list, Opie is BEEKISMA.

Have you checked with the sexton of Herbster Cemetery? There a lot of BEEKSMA burials there.

Onne (1875 - 1956) & Sybil (1875 - 1961)
Rosa (1881 - 1959)
Merle (1898 - 1986) [3 Feb. 1898 - Nov. 1986, per SSDI]
Bokkie (1906 - 1979)
Jake (1907 - 1984) [10 Dec. 1907 - Oct. 1984]
Thomas [7 Apr. 1909 - 27 Je. 1999]
Minnie (1911 - 2000)
Mabel (1912 - 2003) [12 Jan. 1912 - 1 Apr. 2003, lr. Washburn, per SSDI]
Onne (1913 - 1996) [27 July 1913 - 28 Dec. 1996, lr. Iron River]
Opie (5 Sep. 1914 - 12 Feb. 1992)
Edith (1916 - _ )
Esther (1919 - )
Lillian (1918 - 2002)
Lillian (1919 - 1996)
--others more recent.

Is your family connected to Bokka BEEKSMA (1864 - 7 Jan. 1943, Oak Harbor, Island, WA) & wf. Adriana[DEHAAN] (May 1875 - 6 Aug. 1941, Anacortes, Skagit, WA)? Pilot.FamilySearch.org has their death records, showing that both were born in Holland. Adriana was daughter of Siebren DEHAAN & Adriana KUIPERS. Bokka & Adriana had children on the 1920 Census, Island, WA: Siebren, Gertie, Egbert, Bessie, & Jessie.

--not my family