The average transverse momentum ⟨pT⟩ versus the charged-particle multiplicity Nch was measured in p–Pb collisions at a collision energy per nucleon–nucleon pair √sNN = 5.02 TeV and in pp collisions at collision energy of √s=0.9, 2.76 and 7 TeV in a kinematic range 0.15<pT <10.0 Gev/c and |η| < 0.3 with the ALICE apparatus at the LHC. These data are compared to results in Pb–Pb collisions at √sNN = 2.76 TeV at similar charged-particle multiplicities. In pp and p–Pb collisions, a strong increase of ⟨pT⟩ with Nch is observed, which is much stronger than that measured in Pb–Pb collisions. For pp collisions, this could be attributed, within a model of hadronizing strings, to multiple-parton interactions and to a final-state color reconnection mechanism. The data in p–Pb and Pb–Pb collisions cannot be described by an incoherent superposition of nucleon–nucleon collisions and pose a challenge to most of the event generators.