We present the results of the search for gravitational waves (GWs) associated with γ-ray bursts detected duringthe first observing run of the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO). We findno evidence of a GW signal for any of the 41 γ-ray bursts for which LIGO data are available with sufficientduration. For all γ-ray bursts, we place lower bounds on the distance to the source using the optimisticassumption that GWs with an energy of 10-2M c2e were emitted within the 16–500 Hz band, and we find amedian 90% confidence limit of 71 Mpc at 150 Hz. For the subset of 19 short/hard γ-ray bursts, we place lowerbounds on distance with a median 90% confidence limit of 90 Mpc for binary neutron star (BNS) coalescences,and 150 and 139 Mpc for neutron star–black hole coalescences with spins aligned to the orbital angularmomentum and in a generic configuration, respectively. These are the highest distance limits ever achieved byGW searches. We also discuss in detail the results of the search for GWs associated with GRB 150906B, anevent that was localized by the InterPlanetary Network near the local galaxy NGC 3313, which is at aluminosity distance of 54 Mpc (z = 0.0124). Assuming the γ-ray emission is beamed with a jet half-openingangle 30°, we exclude a BNS and a neutron star–black hole in NGC 3313 as the progenitor of this event withconfidence >99%. Further, we exclude such progenitors up to a distance of 102 Mpc and 170 Mpc,respectively.