  We present $^{57}$Fe-NMR measurements of the novel normal and
superconducting-state characteristics of the iron-arsenide superconductor
Ba$_{0.6}$K$_{0.4}$Fe$_2$As$_2$ ($T_c$ = 38 K). In the normal state, the
measured Knight shift and nuclear spin-lattice relaxation rate $(1/T_1)$
demonstrate the development of wave-number ($q$)-dependent spin fluctuations,
except at $q$ = 0, which may originate from the nesting across the disconnected
Fermi surfaces. In the superconducting state, the spin component in the
$^{57}$Fe-Knight shift decreases to almost zero at low temperatures, evidencing
a spin-singlet superconducting state. The $^{57}$Fe-$1/T_1$ results are totally
consistent with a $s^\pm$-wave model with multiple full gaps, regardless of
doping with either electrons or holes.
