Spectroscopic evidence is presented for the existence of the electron-dressed exciton (X(-))-an electron bound to an exciton-in the 8-nm-wide quantum well of an asymmetric n-type AlAs/GaAs triple-barrier resonant tunnelling device. This is revealed by a transition just below the lowest heavy-hole exciton, leading to a doublet structure in the photoluminescence peak. The splitting of 2.1 meV is only weakly dependent on electrical bias. The effect is observed in the neighborhood of each of the three electron current resonances, when the number of electrons in the well is still low but significantly greater than that of photocreated holes. The influence on the X(-) spectrum of applied electrical bias, temperature, excitation photon energy, and excitation power density is investigated. Indications were also found for the existence of X(-) in the narrower, 6-nm-wide quantum well.