Is $\mathbb{A}\setminus0$ an affine scheme for non algebraically closed field

Question

If $k$ is algebraically closed, it's known that $\mathbb{A}^2\setminus0$ is not an affine scheme (see, e.g. this). But what about the scenario when $k$ is not algebraically closed?
My thought: in the case of alg. closed fields, the argument works in the spirit of "there is no polynomial in two variables that vanishes only at $(0,0)$, therefore, any rational function on $\mathbb{A}^2\setminus0$ extends to $\mathbb{A}^2$", but for example, for $k = \mathbb{R}$ there is a polynomial $x ^ 2 + y ^2$ and it turns out that $1/(x^2+y^2)$ does not extend to the whole plane.