  We present high-sensitivity radio continuum observations with the VLA and
Ryle Telescope at 1.5, 4.9, 8.4 and 15.4 GHz of the dwarf irregular galaxy NGC
1569. The radio data show an extended, irregularly-shaped halo with filamentary
structure around the galaxy. The spectral index maps reveal an unusually patchy
distribution with regions of flat spectral index extending into the halo. The
data allow us to perform a spatially-resolved spectral fitting analysis of the
continuum emission from which we derive maps of the thermal and synchrotron
emission. The thermal radio emission is concentrated towards the brightest \hii
region west of the super star clusters A and B whereas the distribution of the
synchrotron emission peaks in a bar-like structure in the disk extending
between the two clusters. The total flux density of the thermal radio emission
allows us to derive the integrated synchrotron spectrum and we confirm the
break in the spectrum that was found by \citet{isr-deb88}. We discuss various
possibilities that could produce such a break and conclude that the only
mechanism able to fit the radio data and being consistent with data at other
wavelengths is a convective wind allowing cosmic ray electrons to escape from
the halo.
