In this Letter, the first evidence of the _{Λ[over ¯]}^{4}He[over ¯] antihypernucleus is presented, along with the first measurement at the LHC of the production of (anti)hypernuclei with mass number A=4, specifically (anti)_{Λ}^{4}H and (anti)_{Λ}^{4}He. In addition, the antiparticle-to-particle ratios for both hypernuclei (_{Λ[over ¯]}^{4}H[over ¯]/_{Λ}^{4}H and _{Λ[over ¯]}^{4}He[over ¯]/_{Λ}^{4}He) are shown, which are sensitive to the baryochemical potential of the strongly interacting matter created in heavy-ion collisions. The results are obtained from a data sample of central Pb-Pb collisions, collected during the 2018 LHC data taking at a center-of-mass energy per nucleon pair of sqrt[s_{NN}]=5.02 TeV. The yields measured for the average of the charge-conjugated states are found to be [0.78±0.19(stat)±0.17(syst)]×10^{-6} for the (anti)_{Λ}^{4}H and [1.08±0.34(stat)±0.20(syst)]×10^{-6} for the (anti)_{Λ}^{4}He, and the measured antiparticle-to-particle ratios are in agreement with unity. The presence of (anti)_{Λ}^{4}H and (anti)_{Λ}^{4}He excited states is expected to strongly enhance the production yield of these hypernuclei. The yield values exhibit a combined deviation of 3.3σ from the theoretical ground-state-only expectation, while the inclusion of the excited states in the calculations leads to an agreement within 0.6σ with the present measurements. Additionally, the measured (anti)_{Λ}^{4}H and (anti)_{Λ}^{4}He masses are compatible with the world-average values within the uncertainties.