  We construct the most general parity-even higher-derivative N=1 off-shell
supergravity model in three dimensions with a maximum of six derivatives.
Excluding terms quadratic in the curvature tensor with two explicit derivatives
and requiring the absence of ghosts in a linearized approximation around an
AdS_3 background, we find that there is a unique supersymmetric invariant which
we call supersymmetric `cubic extended' New Massive Gravity. The purely
gravitational part of this invariant is in agreement with an earlier analysis
based upon the holographic c-theorem and coincides with an expansion of
Born-Infeld gravity to the required order.
  Our results lead us to propose an expression for the bosonic part of
off-shell N=1 Born-Infeld supergravity in three dimensions that is free of
ghosts. We show that different truncations of a perturbative expansion of this
expression gives rise to the bosonic part of (i) Einstein supergravity; (ii)
supersymmetric New Massive Gravity and (iii) supersymmetric `cubic extended'
New Massive Gravity.
