  In this work we first discuss systematically three general approaches to
construct a non-Hermitian flat band, defined by its dispersionless real part.
They resort to, respectively, spontaneous restoration of non-Hermitian
particle-hole symmetry, a persisting flat band from the underlying Hermitian
system, and a compact Wannier function that is an eigenstate of the entire
system. For the last approach in particular, we show the simplest lattice
structure where it can be applied, and we further identify a special case of
such a flat band where every point in the Brillouin zone is an exceptional
point of order 3. A localized excitation in this "EP3 flat band" can display
either a conserved power, quadratic power increase, or even quartic power
increase, depending on whether the localized eigenstate or one of the two
generalized eigenvectors is initially excited. Nevertheless, the asymptotic
wave function in the long time limit is always given by the eigenstate, and in
this case, the compact Wannier function or its superposition in two or more
unit cells.
