Are there $\| \cdot \|_{\infty}$ Kernels?

Question

Some kernels used in machine learning are linked to metrics via the negative exponential function $f(t) = e^{-t^p}$. The most prominent example is the Gaussian RBF kernel $$K(x,y) = e^{-\sigma^2 \|x-y\|_2^2}$$ The connection between metrics and PD kernels was also discussed here. Furthermore, it was shown here that $$K(x,y) = e^{-\|x-y\|_{\infty}}$$ is not PD. However, I am wondering whether there are other ways to connect the maximum norm with some function $f$ such that $f(\|x-y\|_{\infty})$ is actually a PD kernel on $\mathbb{R}^d$.