How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Question

Verifying that (p=2) satisfies

$$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall v\in\mathbb{R}^{n}.\left\|A^kv\right\|_p\!\!=\left\|v\right\|_p$$

is trivial (let each $A$ be a rotation matrix), but is there a clean way to obtain the value two from this condition?

(Note that $A^k$ is ambiguous for some choices of $A$ and $k$. I believe that the solution set is the same regardless of how this ambiguity is resolved, even if such an $A^k$ is taken to be undefined.)

Related, but not (I believe) duplicates:

• Why do we use the Euclidean metric on $\mathbb{R}^2$?
• Why is the 2 norm “special”?
• Why is the Euclidean metric the natural choice?

Probably related:

• This Wikipedia section