A search is presented for a third-generation leptoquark (LQ) coupled exclusively to a τ lepton and a b quark. The search is based on proton-proton collision data at a center- of-mass energy of 13 TeV recorded with the CMS detector, corresponding to an integrated luminosity of 138 fb−1. Events with τ leptons and a varying number of jets originating from b quarks are considered, targeting the single and pair production of LQs, as well as nonresonant t-channel LQ exchange. An excess is observed in the data with respect to the background expectation in the combined analysis of all search regions. For a benchmark LQ mass of 2 TeV and an LQ-b-τ coupling strength of 2.5, the excess reaches a local significance of up to 2.8 standard deviations. Upper limits at the 95% confidence level are placed on the LQ production cross section in the LQ mass range 0.5–2.3 TeV, and up to 3 TeV for t-channel LQ exchange. Leptoquarks are excluded below masses of 1.22–1.88 TeV for different LQ models and varying coupling strengths up to 2.5. The study of nonresonant ττ production through t-channel LQ exchange allows lower limits on the LQ mass of up to 2.3 TeV to be obtained.