Successfully navigating in physical or semantic space requires a neural representation of allocentric (map-based) vectors to boundaries, objects and goals. Cognitive processes such as path-planning and imagination entail the recall of vector representations, but evidence of neuron-level memory for allocentric vectors has been lacking. Here, we describe a novel neuron type, vector trace cell (VTC), whose firing generates a new vector field when a cue is encountered and a 'trace' version of that field for hours after cue removal. VTCs are concentrated in subiculum, distal to CA1. Compared to non-trace cells, VTCs fire at further distances from cues and exhibit earlier-going shifts in preferred theta phase in response to newly introduced cues, which demonstrates a theta-linked neural substrate for memory encoding. VTCs suggest a vector-based model of computing spatial relationships between an agent and multiple spatial objects, or between different objects, freed from the constraints of direct perception of those objects. Poulter et al. report on vector trace cells (VTCs) in the hippocampal subiculum. VTCs support vector coding for previously encountered, now absent, objects and boundaries, potentially facilitating navigation to remembered goals.