The standard Reuleaux triangle is not smooth, but the three points of tangential discontinuity can be smoothed as in the figure below (left), from the Wikipedia article. However, it is unclear (to me) from this diagram whether the curve is $C^2$ or $C^\infty$.

*Meissner’s tetrahedron* is a 3D body of constant width,
but it is not smooth, as is evident in the right figure
below.

My question is:

Are there $C^\infty$ constant-width bodies in $\mathbb{R}^d$ (other than the spheres)?

The image of Meissner’s tetrahedron above is taken
from the impressive work of
Thomas Lachand–Robert and Edouard Oudet,
"Bodies of constant width in arbitrary dimension"
(*Math. Nachr.* 280, No. 7, 740-750 (2007); pre-publication PDF here).

I suspect the answer to my question is known, in which case a reference would suffice. Thanks!

**Addendum.**
Thanks to the knowledgeable (and rapid!) answers by Gerry, Anton, and Andrey,
my question is completely answered—I am grateful!!