Are there any smooth manifolds $M$ with the following property:

There exist a realizing metric $d$ (i.e $d$ induces the topology on $M$), and $d$ is smooth on all of $M \times M$?

If not, is it possibe to guarantee smoothness of the function $x \mapsto d(x,y)$ (for a fixed $y$ ), or smoothness of $d^2$ even on a compact manifold?

(I am trying to see if we can achieve "improved smoothness" if we do not force the metric to be Riemannian.)

Of course, such a metric cannot be induced by a Riemannian metric. (see here and here).