Denote $D=\{x^2+y^2\le1\}\subset\mathbb R^2$ a disk.

Let $f:D\to\mathbb R$ be a function on it. I am interested in restrictions of simple Morse functions on $\mathbb R^2$, but I suspect the answer is the same for any continuous or any smooth function.

A contour of $f$ is a connected component of the preimage of a point, $f^{-1}(\mathrm{const})$.

The diameter of a set is the maximum distance between its points.

Can the maximum diameter of a contour be made arbitrarily small by a suitable choice of $f$?

I suspect that no; namely, I suspect that at least one contour of $f$ will have diameter greater than, say, the radius of the disk, but I can't prove it.

The diameter of the disk is not the bound: Consider a Y-shaped figure formed by three radii, and $f$ being the distance from this figure. Then the largest contour is the Y figure, with the diameter greater than the radius of the disk but smaller than the diameter of the disk.