  We examine an Unruh-DeWitt particle detector coupled to a scalar field in
three-dimensional curved spacetime, within first-order perturbation theory. We
first obtain a causal and manifestly regular expression for the instantaneous
transition rate in an arbitrary Hadamard state. We then specialise to the
Ba\~nados-Teitelboim-Zanelli black hole and to a massless conformally coupled
field in the Hartle-Hawking vacuum. A co-rotating detector responds thermally
in the expected local Hawking temperature, while a freely-falling detector
shows no evidence of thermality in regimes that we are able to probe, not even
far from the horizon. The boundary condition at the asymptotically anti-de
Sitter infinity has a significant effect on the transition rate.
