I forwarded this question to Detlev Hoffmann, who says that such examples exist. Specifically, you can produce such an example where there is, say, an anisotropic form of dimension 8 using characteristic 2 analogues of the techniques in Merkurjev's 1992 article "Simple algebras and quadratic forms". He says details can be found in the PhD thesis of his student Frederic Faivre at Université Franche-Comté.

Detlev further explained the source of Arf's confusion, which I will now recap. Over a field $F$ of any characteristic, the tensor product of two quaternion algebras is not a division algebra if and only if the two quaternion algebras contain a common quadratic extension. In characteristic different from 2, this is an essentially complete criterion. But in characteristic 2, one has to wonder if this quadratic extension is separable or inseparable.

Draxl showed that if two quaternion algebras contain a common quadratic extension, you can always find one that is separable. That is, the property of containing a common inseparable extension is much stronger (because it implies that they contain a common separable extension). A nice exposition of this can be found in T.Y. Lam's 2002 article "On the linkage of quaternion algebras".

Detlev asserts: If you demand that every pair of quaternion division algebras over F (of characteristic 2) share an inseparable quadratic extension, then there are no anisotropic regular quadratic forms of dimension > 4. "This is essentially due to Baeza. In fact, in some sense even to Arf, except that he didn't realize there's a difference between [sharing separable and inseparable subfields]." So presumably this is what Arf was claiming to have proved.

In contrast to this, the requirement that the quaternion algebras form a subgroup of the Brauer group is just that every pair of quaternion division algebras share a separable quadratic extension. This is a much weaker hypothesis, and allows for examples of fields like in Faivre's thesis.

I forwarded this question to Detlev Hoffmann, who says that such examples exist. Specifically, you can produce such an example where there is, say, an anisotropic form of dimension 8 using characteristic 2 analogues of the techniques in Merkurjev's 1992 article "Simple algebras and quadratic forms". He says details can be found in the PhD thesis of his student Frederic Faivre at Université Franche-Comté.

Detlev further explained the source of Arf's confusion, which I will now recap. Over a field $F$ of any characteristic, the tensor product of two quaternion algebras is not a division algebra if and only if the two quaternion algebras contain a common quadratic extension. In characteristic different from 2, this is an essentially complete criterion. But in characteristic 2, one has to wonder if this quadratic extension is separable or inseparable.

Draxl showed that if two quaternion algebras contain a common quadratic extension, you can always find one that is separable. That is, the property of containing a common inseparable extension is much stronger (because it implies that they contain a common separable extension). A nice exposition of this can be found in T.Y. Lam's 2002 article "On the linkage of quaternion algebras".

Detlev asserts: If you demand that every pair of quaternion division algebras over $F$ (of characteristic 2) share an inseparable quadratic extension, then there are no anisotropic regular quadratic forms of dimension > 4. "This is essentially due to Baeza. In fact, in some sense even to Arf, except that he didn't realize there's a difference between [sharing separable and inseparable subfields]." So presumably this is what Arf was claiming to have proved.

In contrast to this, the requirement that the quaternion algebras form a subgroup of the Brauer group is just that every pair of quaternion division algebras share a separable quadratic extension. This is a much weaker hypothesis, and allows for examples of fields like in Faivre's thesis.

Here are some precise references for the theorem "every pair of quaternion algebras over $F$ share an inseparable quadratic extension implies every regular quadratic form of dimension > 4 is isotropic":

• Arf (with confusion mentioned above): Satz 11 in "Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I." J. reine angew. Math. 183, 148-167 (1941)"
• Baeza: Theorem 3.1 in "Comparing $u$-invariants of fields of characteristic $2$." Bol. Soc. Brasil. Mat. 13 (1982), no. 1, 105--114.
• Faivre's thesis: Proposition 3.3.5 (with complete proof)