In this paper we present two results in ($2+1$) gravity coupled to nonlinear electrodynamics. First we determine the general form of the electromagnetic field tensor in ($2+1$) gravity coupled to nonlinear electrodynamics in stationary cyclic spacetimes. Secondly, we determine a family of exact solutions in ($2+1$) gravity sourced by a nonlinear electromagnetic field. The solutions are characterized by five parameters: mass $M$, angular momentum $J$, cosmological constant $\mathrm{\ensuremath{\Lambda}}$, and two electromagnetic charges ${q}_{\ensuremath{\alpha}}$ and ${q}_{\ensuremath{\beta}}$. Remarkably, the solution can be interpreted as a traversable wormhole, provided the fulfillment of certain inequalities by the characteristic parameters; fine-tuning of the cosmological constant leads to an extreme black hole, whereas by switching off one of the electromagnetic charges, we obtain the Ba\~nados-Teitelboim-Zanelli (BTZ) black hole.