PTF 09dav is a peculiar subluminous Type Ia supernova (SN) discovered by the Palomar Transient Factory (PTF). Spectroscopically, it appears superficially similar to the class of subluminous SN1991bg-like SNe, but it has several unusual features which make it stand out from this population. Its peak luminosity is fainter than any previously discovered SN1991bg-like SN Ia (M$_B$ åisebox-0.5ex -15.5), but without the unusually red optical colors expected if the faint luminosity were due to extinction. The photospheric optical spectra have very unusual strong lines of Sc II and Mg I, with possible Sr II, together with stronger than average Ti II and low velocities of i̊sebox-0.5ex 6000 km s$^-1$. The host galaxy of PTF09dav is ambiguous. The SN lies either on the extreme outskirts (rs̊ebox-0.5ex 41 kpc) of a spiral galaxy or in an very faint (M$_R$ >= -12.8) dwarf galaxy, unlike other 1991bg-like SNe which are invariably associated with massive, old stellar populations. PTF 09dav is also an outlier on the light-curve-width-luminosity and color-luminosity relations derived for other subluminous SNe Ia. The inferred $^56$Ni mass is small (0.019 ± 0.003 M $_sun$), as is the estimated ejecta mass of 0.36 M $_sun$. Taken together, these properties make PTF 09dav a remarkable event. We discuss various physical models that could explain PTF 09dav. Helium shell detonation or deflagration on the surface of a CO white dwarf can explain some of the features of PTF 09dav, including the presence of Sc and the low photospheric velocities, but the observed Si and Mg are not predicted to be very abundant in these models. We conclude that no single model is currently capable of explaining all of the observed signatures of PTF 09dav.