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We study extended shift symmetries that arise for fermionic fields on anti-de
Sitter (AdS) space and de Sitter (dS) space for particular values of the mass
relative to the curvature scale. We classify these symmetries for general
mixed-symmetry fermionic fields in arbitrary dimension and describe how fields
with these symmetries arise as the decoupled longitudinal modes of massive
fermions as they approach partially massless points. For the particular case of
AdS$_4$, we look for non-trivial Lie superalgebras that can underly interacting
theories that involve these fields. We study from this perspective the minimal
such theory, the Akulov--Volkov theory on AdS$_4$, which is a non-linear theory
of a spin-$1/2$ Goldstino field that describes the spontaneous breaking of
${\cal N}=1$ supersymmetry on AdS$_4$ down to the isometries of AdS$_4$. We
show how to write the nonlinear supersymmetry transformation for this theory
using the fermionic ambient space formalism. We also study the Lie
superalgebras of candidate multi-field examples and rule out the existence of a
supersymmetric special galileon on AdS$_4$.