We present the discovery of seven new proplyds (i.e., sources surrounded by cometary H alpha emission characteristic of offset ionization fronts (IFs)) in NGC 1977, located about 30' north of the Orion Nebula Cluster (ONC) at a distance of similar to 400 pc. Each of these proplyds is situated at projected distances 0.04-0.27 pc from the B1V star 42 Orionis (c Ori), which is the main source of UV photons in the region. In all cases the IFs of the proplyds are clearly pointing toward the common ionizing source, 42 Ori, and six of the seven proplyds clearly show tails pointing away from it. These are the first proplyds to be found around a B star, with previously known examples instead being located around O stars, including those in the ONC around theta(1) Ori C. The radii of the offset IFs in our proplyds are between similar to 200 and 550 au; two objects also contain clearly resolved central sources that we associate with disks of radii 50-70 au. The estimated strength of the FUV radiation field impinging on the proplyds is around 10-30 times less than that incident on the classic proplyds in the ONC. We show that the observed proplyd sizes are however consistent with recent models for FUV photoevaporation in relatively weak FUV radiation fields.