  In this paper we examine to what extent the radio continuum can be used as an
extinction free probe of star formation in dwarf galaxies. To that aim we
observe $40$ nearby dwarf galaxies with the Very Large Array at 6 cm ($4$-$8$
GHz) in C-configuration. We obtained images with $3$-$8^{\prime\prime}$
resolution and noise levels of $3$-$15{\rm \,\mu Jy\,beam^{-1}}$. We detected
emission associated with $22$ of the $40$ dwarf galaxies, $8$ of which are new
detections. The general picture is that of an interstellar medium largely
devoid of radio continuum emission, interspersed by isolated pockets of
emission associated with star formation. We find an average thermal fraction of
$\sim 50$-$70$% and an average magnetic field strength of $\sim 5$-$8\,{\rm \mu
G}$, only slightly lower than that found in larger, spiral galaxies. At 100 pc
scales, we find surprisingly high values for the average magnetic field
strength of up to 50$\,{\rm \mu G}$. We find that dwarf galaxies follow the
theoretical predictions of the radio continuum-star formation rate relation
within regions of significant radio continuum emission but that the non-thermal
radio continuum is suppressed relative to the star formation rate when
considering the entire optical disk. We examine the far-infrared-star formation
rate relation for our sample and find that the far-infrared is suppressed
compared to the expected star formation rate. We discuss explanations for these
observed relations and the impact of our findings on the radio
continuum-far-infrared relation. We conclude that radio continuum emission at
centimetre wavelengths has the promise of being a largely extinction-free star
formation rate indicator. We find that star formation rates of gas rich, low
mass galaxies can be estimated with an uncertainty of $\pm 0.2$ dex between the
values of $2 \times 10^{-4}$ and $0.1 {\rm M_\odot\,yr^{-1}}$.
