  We prove groups acting cocompactly on locally finite trees with hyperbolic
vertex stabilisers are asynchronously automatic. Combining this with previous
work of the authors we obtain an example of a group satisfying several
non-positive curvature properties (being a $\mathrm{CAT}(0)$ group, an
injective group, a hierarchically hyperbolic group, and having quadratic Dehn
function) which is asynchronously automatic but not biautomatic.
