Solid-state lithium batteries cannot achieve reasonable power densities because of dendrites, whose formation mechanisms remain uncertain. This paper applies principles of chemomechanics to investigate the critical current above which dendrites form in lithium-conductive ceramics. Applied voltage induces stress in solid electrolytes; dendrites appear to nucleate in the exemplary garnet-oxide material Li7La3Zr2O12 (LLZO) when the interfacial pressure exceeds a particular value. The critical pressure of polycrystalline LLZO correlates well with the surface-energy changes incurred by lithium plating in its grain boundaries. A derived formula, validated by experiments, predicts quantitatively how critical current varies with properties including interfacial impedance, bulk permittivity, and grain size. As well as suggesting novel strategies to create more resilient ion-conductive ceramics, the proposed mechanism rationalizes experimental observations of bulk lithium plating and explains how LLZO exhibits an electrically activated transition from stable low-current cyclability to high-current dendrite nucleation.