Is there a gapless Hamiltonian whose gaplessness is stable against arbitrary perturbations?

The answer seems to be: yes.

For example, in an article by Hermele, Fisher and Balents (2003), they consider a $U(1)$ spin liquid in two dimensions where they concretely claim that ``[t]his state is stable to ALL zero-temperature perturbations''.

It is however not clear to me whether this is a generic property of $U(1)$ spin liquids. Often people seem to study such models at high-symmetry points (e.g. in spin systems with $SU(2)$ symmetry), but I do not know whether that is merely to make the solution easier, or whether it is a necessity in those cases.

Stable gapless phases have also been considered in three dimensions, recently (for example) by Rasmussen, You and Xu (2016). In the introduction of that paper, they also give a nice list of references to other works which have considered stable gapless phases (in two and three dimensions).

(Sidenote: it would be interesting to prove such stable gapless phases are not possible in one dimension.)

EDIT: it turns out there are even examples in one spatial dimension! There are so-called 'perfect metals' which are stable against any perturbation (thanks to Ryan Thorngren for pointing this out to me). These require large central charge by virtue of Hellerman's theorem, which says there is always an operator with dimension $\Delta \lessapprox \frac{c}{6} + 0.473695$, hence, there is always a relevant operator if $c \lessapprox 9.13$.