The smooth locus of a *spherical* orbit (i.e. $G(\mathcal O)$-orbit) on the affine Grassmannian is just the spherical orbit itself (but this orbit consists of several Iwahori-orbits). See

Malkin-Ostrik-Vybornov,
The minimal degeneration singularities in the affine Grassmannians,
Duke Math. J. 126 (2005), no. 2, 233–249

for a more precise result: they describe the singularity of each orbit closure along each spherical orbit open in the boundary. They quote

Evens-Mirković
Characteristic cycles for the loop Grassmannian and nilpotent orbits,
Duke Math. J. 97 (1999), no. 1, 109–126

for the result that the smooth locus of a spherical orbit is just the orbit itself. I have an alternative modular representation-theoretic fun proof using the geometric Satake correspondence (see a preprint of mine from 2008).

So the answer to your question was clearly no as stated on general grounds: for a general Kac-Moody Schubert variety, if w is maximal modulo some finite parabolic subgroup $W_I$, then the smooth locus of $\overline{BwB}/B$ contains $P_IwB$, which is the union of the Schubert cells corresponding to the elements of the coset $W_I w$, so more than one if $W_I$ is nontrivial (for $W$ finite, the case of the full flag variety is $W_I = W$). If one considers $G/P$ instead of $G/B$, I guess one has to consider double cosets...

But the result above tells you that for $\lambda$ dominant, the smooth locus does not get larger than that (considering the finite Weyl group as a maximal parabolic subgroup of the affine Weyl group). I don't know the answer for general $\lambda$, but there is an extensive literature about smooth (resp. rationally smooth, even p-smooth) loci of Schubert varieties.

A similar result holds for nilpotent cones: the smooth locus of a nilpotent orbit is just the orbit itself (this is a consequence of results by Namikawa and Kaledin; again, one can describe all minimal degeneration singularities: this was done by Kraft-Procesi in classical types, and by Fu-J.-Levy-Sommers in exceptional types).