  In this paper we introduce Parity-Time ($\cal PT$) symmetric perturbation to
a one-dimensional Lieb lattice, which is otherwise $\cal P$-symmetric and has a
flat band. In the flat band there are a multitude of degenerate dark states,
and the degeneracy $N$ increases with the system size. We show that the
degeneracy in the flat band is completely lifted due to the non-Hermitian
perturbation in general, but it is partially maintained with the
half-gain-half-loss perturbation and its ``V" variant that we consider. With
these perturbations, we show that both randomly positioned states and pinned
states at the symmetry plane in the flat band can undergo thresholdless $\cal
PT$ breaking. They are distinguished by their different rates of acquiring
non-Hermicity as the $\cal PT$-symmetric perturbation grows, which are
insensitive to the system size. Using a degenerate perturbation theory, we
derive analytically the rate for the pinned states, whose spatial profiles are
also insensitive to the system size. Finally, we find that the presence of weak
disorder has a strong effect on modes in the dispersive bands but not on those
in the flat band. The latter respond in completely different ways to the
growing $\cal PT$-symmetric perturbation, depending on whether they are
randomly positioned or pinned.
