  Rashba interfaces yield efficient spin-charge interconversion and give rise
to nonreciprocal transport phenomena. Here, we report magnetotransport
experiments in few-nanometer-thick films of PdCoO$_2$, a delafossite oxide
known to display a large Rashba splitting and surface ferromagnetism. By
analyzing the angle dependence of the first- and second-harmonic longitudinal
and transverse resistivities, we identify a Rashba-driven unidirectional
magnetoresistance that competes with the anomalous Nernst effect below the
Curie point. We estimate a Rashba coefficient of 0.75 {\pm} 0.3 eV {\AA} and
argue that our results qualify delafossites as a new family of oxides for
nano-spintronics and spin-orbitronics, beyond perovskite materials.
