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Aims. We investigate the long gamma-ray burst (GRB) 140629A through multiwavelength observations to derive the properties of the dominant jet and its host galaxy. Methods. The afterglow and host galaxy observations were taken in the optical (Swift/UVOT and various facilities worldwide), infrared (Spitzer), and X-rays (Swift/XRT) between 40 s and 3 yr after the burst trigger. Results. Polarisation observations by the MASTER telescope indicate that this burst is weakly polarised. The optical spectrum contains absorption features, from which we confirm the redshift of the GRB as originating at z = 2.276 ± 0.001. We performed spectral fitting of the X-rays to optical afterglow data and find there is no strong spectral evolution. We determine the hydrogen column density NH to be 7.2 × 1021 cm−2 along the line of sight. The afterglow in this burst can be explained by a blast wave jet with a long-lasting central engine expanding into a uniform medium in the slow cooling regime. At the end of energy injection, a normal decay phase is observed in both the optical and X-ray bands. An achromatic jet break is also found in the afterglow light curves ∼0.4 d after trigger. We fit the multiwavelength data simultaneously with a model based on a numerical simulation and find that the observations can be explained by a narrow uniform jet in a dense environment with an opening angle of 6.7◦ viewed 3.8◦ off-axis, which released a total energy of 1.4 × 1054 erg. Using the redshift and opening angle, we find GRB 140629A follows both the Ghirlanda and Amati relations. From the peak time of the light curve, identified as the onset of the forward shock (181s after trigger), the initial Lorentz factor (Γ0) is constrained in the range 82-118. Fitting the host galaxy photometry, we find the host to be a low mass, star-forming galaxy with a star formation rate of log (SFR) = 1.1+−00.94 M yr−1. We obtain a value of the neutral hydrogen density by fitting the optical spectrum, log NHI = 21.0 ± 0.3, classifying this host as a damped Lyman-alpha. High ionisation lines (N v, Si iv) are also detected in the spectrum. © ESO 2019. ; China Scholarship Council, CSC ; National Science Foundation, NSF ; National Basic Research Program of China (973 Program): DST/IMRCD/BRICS/Pilotcall/ProFCheap/2017, 2018YFA0404204, NSFC-11833003 ; 17-52-80133 ; Agenzia Spaziale Italiana, ASI: 2015-046-R.0 ; Ministry of Education and Science of the Russian Federation, Minobrnauka: 2019-05-595-0001-2496, 2019-05-592-0001-7293 ; 17-52-80139 BRICS-a ; Agenzia Spaziale Italiana, ASI ; 201406660015 ; Leverhulme Trust ; European Regional Development Fund, FEDER: AYA-2015-71718-R ; National Research Foundation, NRF: 2018R1A2A1A05022685 ; Horizon 2020 Framework Programme, H2020: 654215 ; ★ Research supported by the China Scholarship Council. † Deceased. ; 7 IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation. http://ast.noao.edu/data/software ; Acknowledgements. Acknowledge the support by the program of China Scholarships Council (CSC) under the Grant no. 201406660015. We also acknowledge support from the Spanish MINEICO ministry and European FEDER funds AYA-2015-71718-R. SRO gratefully acknowledges the support of the Leverhulme Trust Early Career Fellowship. RS-R acknowledges support from ASI (Italian Space Agency) through the Contract no. 2015-046-R.0 and from European Union Horizon 2020 Programme under the AHEAD project (grant agreement no. 654215). MASTER equipment is supported by Lomonosov MSU Development Program and by Moscow Union OPTIKA. VL,EG, NT, VK are supported by BRICS RFBR grant 17-52-80133. MASTER-Tunka equipment is supported of Russian Federation Ministry of Science and High Education (grants 2019-05-592-0001-7293 and 2019-05-595-0001-2496). B.-B.Z. acknowledges support from the National Key Research and Development Program of China (2018YFA0404204), and NSFC-11833003. S.B.P. acknowledges BRICS grant DST/IMRCD/BRICS/Pilotcall/ProFCheap/2017(G) for this work. I.D. acknowledges L. Piro his invitation and financial support to visit and work at IAPS (Rome). We also acknowledge the use of the public data from the Swift data archive. We thank the excellent support form the GTC staff which is located at Observatorio del Roque de los Muchachos at Canary Islands (Spain). Thanks to the data support by NASA with Spitzer Space Telescope. SP and RB acknowledge support from RBRF grant 17-52-80139 BRICS-a. IHP acknowledges support from NRF 2018R1A2A1A05022685. Finally, we want to thank the anonymous referee for his/her comments, which have substantially improved the manuscript.