Laser cooling of mechanical degrees of freedom is one of the most significant achievements in the field of cavity optomechanics [1]. Mesoscopic mechanical oscillators with high resonant frequencies (MHz to GHz) are typically favoured for laser cooling because they allow easier access to the sideband-resolved regime. The extension of this technique to the macroscopic scale, which usually involves lower frequency resonances, is not straightforward and several schemes have been proposed over the past decade [2,3,4,5]. Here we report efficient passive optomechanical cooling of the motion of a free-standing waveguide that is dissipatively coupled to a high-Q whispering-gallery-mode (WGM) bottle resonator. The waveguide is an 8 mm long glass-fibre nanospike [6], which has a fundamental mechanical resonance at Ω/2π = 2.5 kHz. Upon launching ~250 μW laser power at an optical frequency close to the WGM resonant frequency, the optomechanical interaction between nanospike and WGM resonator causes the nanospike resonance to be cooled from room temperature down to 1.8 K. Simultaneous cooling of the first higher order mechanical mode is also observed, causing strong suppression of the Brownian motion of the nanospike, observed as an 11.6 dB reduction in its mean square displacement. This result sets a new benchmark on the lowest frequency mechanical motion that can be passively cooled, and represents the first practical application of dissipative optomechanics. The results are of direct relevance in the many applications of high-Q WGM resonators, including nonlinear optics [7], atom physics [8], optomechanics [9, 10], and sensing [11,12].