  We first make a Killing spinor analysis for a general three-dimensional
off-shell $N=(2,0)$ supergravity and find conditions for a bosonic background
to preserve some supersymmetry . We then consider a particular model, namely
$N=(2,0)$ topologically massive supergravity and impose its field equations. By
making a suitable ansatz on metric functions we find a large class of solutions
that include spacelike, timelike and null warped $AdS_3$ among others.
Isometric quotients of spacelike and timelike squashed $AdS_3$ solutions yield
extremal black holes without closed causal curves.
