  Context. The details of cosmic-ray transport have a strong impact on galaxy
evolution. The peak of the cosmic-ray energy distribution is observable in the
radio continuum using the electrons as proxy. Aims. We measure the length that
the cosmic-ray electrons (CRE) are transported during their lifetime in the
nearby galaxy M 51 across one order of magnitude in cosmic-ray energy
(approximately 1-10 GeV). To this end we use new ultra-low frequency
observations from the LOw Frequency ARay (LOFAR) at 54 MHz and ancillary data
between 144 and 8350 MHz. Methods. As the the CRE originate from supernova
remnants, the radio maps are smoothed in comparison to the distribution of the
star formation. By convolving the map of the star-formation rate (SFR) surface
density with a Gaussian kernel, we can linearise the radio-SFR relation. The
best-fitting convolution kernel is then our estimate of the CRE transport
length. Results. We find that the CRE transport length increases at low
frequencies, as expected since the CRE have longer lifetimes. The CRE transport
length is $l_{\rm CRE} = \sqrt{4Dt_{\rm syn}}$, where $D$ is the isotropic
diffusion coefficient and $t_{\rm syn}$ is the CRE lifetime as given by
synchrotron and inverse Compton losses. We find that the data can be well
fitted by diffusion, where $D=(2.14\pm 0.13) \times 10^{28}~\rm cm^2\,s^{-1}$.
With $D\propto E^{0.001\pm 0.185}$, the diffusion coefficient is independent of
the CRE energy $E$ in the range considered. Conclusions. Our results suggest
that the transport of GeV-cosmic ray electrons in the star-forming discs of
galaxies is governed by energy-independent diffusion.
