The answer to the title question is yes (well, I assume that by a "surface" you mean something reasonable, like a boundary of a convex set).

Let $AB$ be the longest segment with endpoints on the surface. We may assume that its length equals 2 and its midpoint is the origin. Consider projections to the planes that contain $AB$. Since projections do not increase distances, $AB$ is a diameter of each projection. Hence all projections to this family of planes are unit discs centered at the origin. The intersection of the corresponding cylinders is the unit ball, hence the result.