Aims. We observed four nearby spiral galaxies (NGC 3627, NGC 4254, NGC 4736, and NGC 5055) in the K band with the 64-m Sardinia Radio Telescope, with the aim of detecting anomalous microwave emission (AME), a radiation component presumably due to spinning dust grains, which has been observed thus far in the Milky Way and only in a handful of other galaxies (most notably, M 31). Methods. We mapped the galaxies at 18.6 and 24.6 GHz and studied their global photometry together with other radio-continuum data from the literature in order to find AME as emission in excess of the synchrotron and thermal components. Results. We only found upper limits for AME. These nondetections, and other upper limits in the literature, are nevertheless consistent with the average AME emissivity from a few detections: it is ϵ_{30 GHz}^AME = 2.4 ± 0.4 × 10⁻² MJy sr⁻¹ (M_⊙ pc⁻²)⁻¹ in units of dust surface density (equivalently, 1.4 ± 0.2 × 10⁻¹⁸ Jy sr⁻¹ (H cm⁻²)⁻¹ in units of H column density). We finally suggest searching for AME in quiescent spirals with relatively low radio luminosity, such as M 31.