  We show that there exist supersymmetric Lifshitz vacua in off-shell
Einstein-Weyl supergravity, in addition to the BPS AdS_4 vacuum. The Lifshitz
exponents are determined by the product of the cosmological constant and the
coupling of the Weyl-squared term. We then obtain the equations of the
supersymmetric solutions that are asymptotic to the AdS or Lifshitz vacua. We
obtain many examples of exact solutions as well as numerical ones. We find
examples of extremal AdS black branes whose near-horizon geometry is
AdS_2\times T^2. We also find an extremal Lifshitz black hole with z=-2, whose
horizon coincides with the curvature singularity. However the asymptotic
Lifshitz solutions are in general smooth wormholes. In conformal supergravity,
we find intriguing examples of non-extremal "BPS" AdS and Lifshitz black holes
whose local Killing spinor is divergent on the horizon. We show that all the
supersymmetric asymptotic AdS and Lifshitz solutions have the vanishing Noether
charge associated with some scaling symmetry. We also study the integrability
condition of the Killing spinor equation and the supersymmetric invariance of
the action. Finally we show that the only spherically-symmetric BPS solution is
the AdS vacuum.
