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The yields of strange (K0S, Λ, Λ¯) and multi-strange (Ξ−, Ξ¯+, Ω−, Ω¯+) hadrons are measured at midrapidity in proton-proton (pp) collisions at s√ = 7 TeV as a function of the charged-particle multiplicity density (dNch/dη).The production rate of strange particles increases faster than that of non-strange hadrons, leading to an enhancement of strange particles relative to pions, similar to that found in nucleus-nucleus collisions as well as in proton-nucleus collisions at the LHC. This is the first observation of an enhanced production of strange particles in high-multiplicity pp collisions. The magnitude of this strangeness enhancement increases with the event activity, quantified by dNch/dη, and with hadron strangeness. It reaches almost a factor of two for the Ω at the highest multiplicity presented. No enhancement is observed for particles with no strange quark content, demonstrating that the observed effect is strangeness rather than mass related. The results are not reproduced by any of the Monte Carlo models commonly used at the LHC, suggesting that further developments are needed for a complete microscopic understanding of strangeness production and indicating the presence of a phenomenon novel in high-multiplicity pp collisions.