  Quantum fluctuations originating phase competition or geometrical frustration
of spins lead to novel states such as a quantum critical point and a quantum
spin liquid where the strong quantum fluctuations suppress any ordered states
even at 0 K. Utilizing site-selective NMR for a quasi-two dimensional organic
conductor $\lambda$-(STF)$_2$GaCl$_4$, we investigate the non-magnetic
insulating phase of the stripe lattice system consisting of triangular and
square tilings. We found development of AF spin fluctuations with decreasing
temperature. Regardless of large enhancement of spin-lattice relaxation rate
$1/T_1$ owing to critical slowing down below 10 K, no long-range magnetic
ordering was observed down to 1.63 K two orders of magnitude less than the
exchange interaction $J/k_{\rm B} \simeq$ 194 K. Moreover, $1/T_1$ saturated
below 3.5 K. These results are in stark contrast to observed behaviors so far
in other non-magnetic ground states discussed in terms of spin liquids,
demonstrating realization of an exotic quantum state accompanying quantum
criticality.
