This work considers the design of a reference governor to satisfy pointwise-in-time output and control constraints in the setting of data-driven control of aerospace systems with unknown input-output dynamics. This unknown dynamics lumps together the combined effects of unknown internal (state space) dynamics, disturbance forces and torques, and unknown internal (mass/inertia) parameters. The unknown dynamics are modeled by a control-affine ultra-local model (ULM) in discrete time. The reference governor is an add-on scheme that enforces the output and control constraints by modifying, when required, a reference command to the system with unknown input-output dynamics. The reference command is determined on the basis of constraint admissible sets constructed in a data-driven setting and exploiting our ULM. A Lyapunov analysis is carried out to ensure that the output of the reference governor-based control system converges to a desired output trajectory that meets the constraints. Numerical simulation results for aircraft longitudinal flight control are reported with this reference governor-based data-driven control scheme, which demonstrate the performance of the controller and the enforcement of the constraints.