We report persistent photocurrent in p-type nitrogen-doped ZnSe epilayers grown by molecular-beam epitaxy on GaAs. It is observed up to room temperature in some samples with a decay time ranging from several minutes to hours. A typical decay consists of an initial stretched-exponential transient and a subsequent slower transient. We demonstrate that the persistent photocurrent has two components, one of them originating from the presence of metastable centers in the ZnSe layer at the heterointerface, the other from tunneling of trapped holes from a two-dimensional quantum well at the heterojunction through its energy barrier.