Using the ZEUS detector at HERA, we have studied the reaction e+p → e+X for Q2 > 5000 GeV2 with a 20.1 pb-1 data sample collected during the years 1994 to 1996. For Q2 below 15000 GeV2, the data are in good agreement with Standard Model expectations. For Q2 > 35000 GeV2, two events are observed while 0.145 ± 0.013 events are expected. A statistical analysis of a large ensemble of simulated Standard Model experiments indicates that with probability 6.0%, an excess at least as unlikely as that observed would occur above some Q2 cut. For x > 0.55 and y > 0.25, four events are observed where 0.91 ± 0.08 events are expected. A statistical analysis of the two-dimensional distribution of the events in x and y yields a probability of 0.72% for the region x > 0.55 and y > 0.25 and a probability of 7.8% for the entire Q2 > 5000 GeV2 data sample. The observed excess above Standard Model expectations is particularly interesting because it occurs in a previously unexplored kinematic region. © Springer-Verlag 1997.