A search for the production of top-quark pairs with the same electric charge (tt or (tt) over bar) is presented. The analysis uses proton-proton collision data at root s = 13 TeV, recorded by the ATLAS detector at the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb(-1). Events with two same-charge leptons and at least two b-tagged jets are selected. Neural networks are employed to define two selections sensitive to additional couplings beyond the Standard Model that would enhance the production rate of same-sign top-quark pairs. No significant signal is observed, leading to an upper limit on the total production cross-section of same-sign top-quark pairs of 1.6 fb at 95% confidence level. Corresponding limits on the three Wilson coefficients associated with the O-tu((1)), O-Qu((1)), and O-Qu((8)) operators in the Standard Model Effective Field Theory framework are derived.