The application of colloidal semiconductor quantum dots as single-dot light sources still requires several challenges to be overcome. Recently, there has been considerable progress in suppressing intensity fluctuations (blinking) by encapsulating an emitting core in a thick protective shell. However, these nanostructures still show considerable fluctuations in both emission energy and linewidth. Here we demonstrate type-I core/shell heterostructures that overcome these deficiencies. They are made by combining wurtzite semiconductors with a large, directionally anisotropic lattice mismatch, which results in strong asymmetric compression of the emitting core. This modifies the structure of band-edge excitonic states and leads to accelerated radiative decay, reduced exciton-phonon interactions, and suppressed coupling to the fluctuating electrostatic environment. As a result, individual asymmetrically strained dots exhibit highly stable emission energy (<1 meV standard deviation) and a subthermal room-temperature linewidth (~20 meV), concurrent with nearly nonblinking behaviour, high emission quantum yields, and a widely tunable emission colour.