We present the first measurement of event-by-event fluctuations in the kaon sector in Pb – Pb collisions at s_NN=2.76 TeV with the ALICE detector at the LHC. The robust fluctuation correlator ν_dyn is used to evaluate the magnitude of fluctuations of the relative yields of neutral and charged kaons, as well as the relative yields of charged kaons, as a function of collision centrality and selected kinematic ranges. While the correlator ν_dyn[K⁺,K⁻] exhibits a scaling approximately in inverse proportion of the charged particle multiplicity, ν_dyn[K_S⁰,K^±] features a significant deviation from such scaling. Within uncertainties, the value of ν_dyn[K_S⁰,K^±] is independent of the selected transverse momentum interval, while it exhibits a pseudorapidity dependence. The results are compared with HIJING, AMPT and EPOS–LHC predictions, and are further discussed in the context of the possible production of disoriented chiral condensates in central Pb – Pb collisions.