We present the first application of a 'multiple-tracer' redshift-space distortion (RSD) analysis to an observational galaxy sample, using data from the Galaxy and Mass Assembly (GAMA) survey. Our data set is an r < 19.8 magnitude-limited sample of 178 579 galaxies covering the redshift interval z < 0.5 and area 180 deg2. We obtain improvements of 10-20 per cent in measurements of the gravitational growth rate compared to a single-tracer analysis, deriving from the correlated sample variance imprinted in the distributions of the overlapping galaxy populations. We present new expressions for the covariances between the auto-power and cross-power spectra of galaxy samples that are valid for a general survey selection function and weighting scheme. We find no evidence for a systematic dependence of the measured growth rate on the galaxy tracer used, justifying the RSD modelling assumptions, and validate our results using mock catalogues from N-body simulations. For multiple tracers selected by galaxy colour, we measure normalized growth rates in two independent redshift bins f...8(z = 0.18) = 0.36 ± 0.09 and f...8(z = 0.38) = 0.44 ± 0.06, in agreement with standard GR gravity and other galaxy surveys at similar redshifts. (ProQuest: ... denotes formulae/symbols omitted.)