  We construct the first examples of non-smoothable self-homeomorphisms of
smooth $4$-manifolds with boundary that fix the boundary and act trivially on
homology. As a corollary, we construct self-diffeomorphisms of $4$-manifolds
with boundary that fix the boundary and act trivially on homology but cannot be
isotoped to any self-diffeomorphism supported in a collar of the boundary and,
in particular, are not isotopic to any generalised Dehn twist.
