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Because overall metallicity becomes lower at higher redshifts, deep near-infrared transient surveys probing high-redshift supernovae are suitable to discover pair-instability supernovae. The <jats:italic>Euclid<\/jats:italic> satellite, which is planned launch in 2023, has a near-infrared wide-field instrument that is suitable for a high-redshift supernova survey. The Euclid Deep Survey is planned to make regular observations of three Euclid Deep Fields (40\u2006deg<jats:sup>2<\/jats:sup> in total) spanning <jats:italic>Euclid<\/jats:italic>\u2019s six-year primary mission period. While the observations of the Euclid Deep Fields are not frequent, we show that the predicted long duration of pair-instability supernovae would allow us to search for high-redshift pair-instability supernovae with the Euclid Deep Survey. Based on the current observational plan of the <jats:italic>Euclid<\/jats:italic> mission, we conduct survey simulations in order to estimate the expected numbers of pair-instability supernova discoveries. We find that up to several hundred pair-instability supernovae at <jats:italic>z<\/jats:italic>\u2004\u2272\u20043.5 can be discovered within the Euclid Deep Survey. We also show that pair-instability supernova candidates can be efficiently identified by their duration and color, which can be determined with the current Euclid Deep Survey plan. We conclude that the <jats:italic>Euclid<\/jats:italic> mission can lead to the first confirmation of pair-instability supernovae if their event rates are as high as those predicted by recent theoretical studies. 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