  Dynamical symmetries are algebraic constraints on quantum dynamical systems,
which are often responsible for persistent temporal periodicity of observables.
In this work, we discuss how an extensive set of strictly local dynamical
symmetries can exist in an interacting many-body quantum system. These strictly
local dynamical symmetries lead to spontaneous breaking of continuous
time-translation symmetry, i.e. the formation of extremely robust and
persistent oscillations when an infinitesimal time-dependent perturbation is
applied to an arbitrary initial (stationary) state. Observables which do not
overlap with the local (dynamical) symmetry operators can relax, losing memory
of their initial conditions. The remaining observables enter highly robust
non-equilibrium limit cycles, signaling the emergence of a non-trivial
\emph{quantum many-body attractor}. We provide an explicit recipe for
constructing Hamiltonians featuring local dynamical symmetries. As an example,
we introduce the XYZ spin-lace model, which is a model of a quasi-1D quantum
magnet.
