  Nonlocality, as established by seminal Bell's theorem, is considered to be
the most striking feature of correlations present in space like separated
events. Its practical application in device independent protocols, such as
secure key distribution, randomness certification, {\it etc.}, demands
identification and amplification of such correlations observed in the quantum
world. In this Letter we study the prospect of nonlocality distillation,
wherein, by applying a natural set of free operations (called wirings) on many
copies of weakly nonlocal systems, one aims to generate correlations of higher
nonlocal strength. In the simplest Bell scenario, we identify a protocol,
namely, logical OR-AND wiring, that can distil nonlocality to significantly
high degree starting from arbitrarily weak quantum nonlocal correlations. As it
turns out, our protocol has several interesting facets: (i) it demonstrates
that set of distillable quantum correlations has non zero measure in the full
eight-dimensional correlation space, (ii) it can distil quantum Hardy
correlations by preserving its structure, (iii) it shows that (nonlocal)
quantum correlations sufficiently close to the local deterministic points can
be distilled by a significant amount. Finally, we also demonstrate efficacy of
the considered distillation protocol in detecting postquantum correlations.
