The anomalous microwave emission detected in the Perseus molecular complex by Watson et al. has been observed at 11 GHz through dual orthogonal polarizations with the COSMOSOMAS experiment. Stokes U and Q maps were obtained at a resolution of ~0.9d for a 30°×30° region including the Perseus molecular complex. Faint polarized emission has been measured; we find Q=-0.2%+/-1.0% and U=-3.4+1.8-1.4%, both at the 95% confidence level, with a systematic uncertainty estimated to be lower than 1% determined from tests of the instrumental performance using unpolarized sources in our map as null hypothesis. The resulting total polarization level is Π=3.4+1.5-1.9%. These are the first constraints on the polarization properties of an anomalous microwave emission source. The low level of polarization seems to indicate that the particles responsible for this emission in the Perseus molecular complex are not significantly aligned in a common direction over the whole region, as a consequence of either a high structural symmetry in the emitting particle or a low-intensity magnetic field. Our weak detection is fully consistent with predictions from electric dipole emission and resonance relaxation at this frequency.