We present the results of the search for gravitational waves (GWs) associated with $gamma$-ray bursts detected during the first observing run of the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO). We find no evidence of a GW signal for any of the 41 $gamma$-ray bursts for which LIGO data are available with sufficient duration. For all $gamma$-ray bursts, we place lower bounds on the distance to the source using the optimistic assumption that GWs with an energy of $10^{-2}M_odot c^2$ were emitted within the $16$-$500,$Hz band, and we find a median 90% confidence limit of 71$,$Mpc at 150$,$Hz. For the subset of 19 short/hard $gamma$-ray bursts, we place lower bounds 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 angular momentum and in a generic configuration, respectively. These are the highest distance limits ever achieved by GW searches. We also discuss in detail the results of the search for GWs associated with GRB 150906B, an event that was localized by the InterPlanetary Network near the local galaxy NGC 3313, which is at a luminosity distance of 54$,$Mpc ($z=0.0124$). Assuming the $gamma$-ray emission is beamed with a jet half-opening angle $leq 30^{circ}$, we exclude a BNS and a neutron star-black hole in NGC 3313 as the progenitor of this event with confidence $>99$%. Further, we exclude such progenitors up to a distance of 102$,$Mpc and 170$,$Mpc, respectively.
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