Graphene nanoribbons are widely regarded as promising building blocks for next-generation carbon-based devices. A critical issue to their prospective applications is whether their electronic structure can be externally controlled. Here, we combine simple model Hamiltonians with extensive first-principles calculations to investigate the response of armchair graphene nanoribbons to transverse electric fields. Such fields can be achieved either upon laterally gating the nanoribbon or incorporating ambipolar chemical codopants along the edges. We reveal that the field induces a semiconductor-to-semimetal transition with the semimetallic phase featuring zero-energy Dirac fermions that propagate along the armchair edges. The transition occurs at critical fields that scale inversely with the width of the nanoribbons. These findings are universal to group-IV honeycomb lattices, including silicene and germanene nanoribbons, irrespective of the type of edge termination. Overall, our results create new opportunities to electrically engineer Dirac semimetallic phases in otherwise semiconducting graphene-like nanoribbons.