A search for non-resonant Higgs boson pair production in the bbbb final state is presented. The analysis uses up to 139 inverse femtobarns of pp collision data at 13 TeV collected with the ATLAS detector. The analysis targets Higgs boson pairs produced via gluon-gluon fusion including a diagram involving the Higgs self-coupling and decaying into four resolved b-tagged jets. The observed data are consistent with Standard Model predictions. A 95% confidence upper limit on the standard model cross-section is set at 284 fb, and the trilinear Higgs self-coupling parameter is constrained to the interval -5.5 and 12.7 times the Standard Model expectation, assuming Standard Model values for all other couplings.