A search for the single production of a vectorlike top partner (Formula Presented) with mass greater than 1 TeV decaying into a Formula Presented boson and a top quark is presented, using the full Run 2 dataset corresponding to Formula Presented of Formula Presented collisions at Formula Presented, collected in 2015-2018 with the ATLAS detector at the Large Hadron Collider. The targeted final state is characterized by the presence of a pair of electrons or muons with opposite-sign charges which form a Formula Presented-boson candidate, as well as by the presence of Formula Presented-tagged jets and forward jets. Events with exactly two or at least three leptons are categorized into two independently optimized analysis channels. No significant excess above the background expectation is observed and the results from the two channels are statistically combined to set exclusion limits at 95% confidence level on the masses and couplings of Formula Presented. The results are interpreted in several benchmark scenarios to set limits on the mass and universal coupling strength (Formula Presented) of the vectorlike quark. For singlet Formula Presented quarks, Formula Presented values between 0.22 and 0.64 are excluded for masses between 1000 and 1975 GeV. For Formula Presented quarks in the doublet scenario, where the production cross section is much lower, Formula Presented values between 0.54 and 0.88 are excluded for masses between 1000 and 1425 GeV.