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We investigate domain wall and other defect solutions in the weak-field limitof Eddington-inspired Born-Infeld gravity as a function of $\kappa$, the onlyadditional parameter of the theory with respect to General Relativity. Wedetermine, both analytically and numerically, the internal structure of domainwalls, quantifying its dependency on $\kappa$ as well as the impact of suchdependency on the value of the tension measured by an outside observer. We findthat the pressure in the direction perpendicular to the domain wall can be, incontrast to the weak-field limit of General Relativity, significantly greateror smaller than zero, depending, respectively, on whether $\kappa$ is positiveor negative. We further show that the generalized von Laue condition, whichstates that the average value of the perpendicular pressure is approximatelyequal to zero in the weak-field limit of General Relativity, does not generallyhold in EiBI gravity not only for domain walls, but also in the case cosmicstrings and spherically symmetric particles. We argue that a violation of thegeneralized von Laue condition should in general be expected in any theory ofgravity whenever geometry plays a significant role in determining the defectstructure.

Domain walls and other defects in Eddington-inspired Born-Infeld gravity