Context. The study of stars that evolve through the asymptotic giant branch (AGB) proves crucial in several astrophysical contexts because these objects provide important feedback to the host system in terms of the gas that is poured into the interstellar medium after being exposed to contamination from nucleosynthesis processes, and in terms of the dust that forms in their wind. Most of the studies conducted so far have been focused on AGB stars with solar and sub-solar chemical composition, whereas the extremely metal-poor domain has been poorly explored. Aims. We study the evolution of extremely metal-poor AGB stars with metallicities down to [Fe/H] = -5 to understand the main evolutionary properties and the efficiency of the processes able to alter their surface chemical composition, and to determine the gas and dust yields. Methods. We calculated two sets of evolutionary sequences of stars in the 1-7.5M(circle dot) mass range that evolved from the pre-main sequence to the end of the AGB phase. To explore the extremely metal-poor chemistries, we adopted the metallicities Z= 3 x 10(-5) and Z= 3 x 10(-7), which correspond to [Fe/H]= -3 and [Fe/H]= -5, respectively. The results from stellar evolution modelling were used to calculate the yields of the individual chemical species. We also modelled dust formation in the wind to determine the dust produced by these objects. Results. The evolution of AGB stars in the extremely metal-poor domain we explored proves highly sensitive to the initial mass of the star. M <= 2 M-circle dot stars experience several third-dredge-up events, which favour the gradual surface enrichment of C-12 and the formation of significant quantities of carbonaceous dust, similar to 0.01 M-circle dot. The C-13 and nitrogen yields are found to be significantly smaller than in previous explorations of low-mass metal-poor AGB stars because the proton ingestion episodes experienced during the initial AGB phases are weaker. M >= 5 M-circle dot stars experience hot bottom burning, and their surface chemistry reflects the equilibria of a very advanced proton-capture nucleosynthesis; little dust production takes place in their wind. Intermediate-mass stars experience both third dredge-up and hot bottom burning: they prove efficient producers of nitrogen, which is formed by proton captures on C-12 nuclei of primary origin dredged up from the internal regions.