  We report $^{75}$As nuclear magnetic resonance (NMR) studies on a new
iron-based superconductor CaKFe$_4$As$_4$ with $T_{\rm c}$ = 35 K. $^{75}$As
NMR spectra show two distinct lines corresponding to the As(1) and As(2) sites
close to the K and Ca layers, respectively, revealing that K and Ca layers are
well ordered without site inversions. We found that nuclear quadrupole
frequencies $\nu_{\rm Q}$ of the As(1) and As(2) sites show an opposite
temperature ($T$) dependence. Nearly $T$ independent behavior of the Knight
shifts $K$ are observed in the normal state, and a sudden decrease in $K$ in
the superconducting (SC) state clearly evidences spin-singlet Cooper pairs.
$^{75}$As spin-lattice relaxation rates 1/$T_1$ show a power law $T$ dependence
with different exponents for the two As sites. The isotropic antiferromagnetic
spin fluctuations characterized by the wavevector ${\bf q}$ = ($\pi$, 0) or (0,
$\pi$) in the single-iron Brillouin zone notation are revealed by 1/$T_1T$ and
$K$ measurements. Such magnetic fluctuations are necessary to explain the
observed temperature dependence of the $^{75}$As quadrupole frequencies, as
evidenced by our first-principles calculations. In the SC state, 1/$T_1$ shows
a rapid decrease below $T_{\rm c}$ without a Hebel-Slichter peak and decreases
exponentially at low $T$, consistent with an $s^{\pm}$ nodeless two-gap
superconductor.
