Quantum photonic technologies such as quantum communication, sensing or computation require efficient, stable and pure single-photon sources. Epitaxial quantum dots (QDs) have been made capable of on-demand photon generation with high purity, indistinguishability and brightness, although they require precise fabrication and face challenges in scalability. By contrast, colloidal QDs are batch synthesized in solution but typically have broader linewidths, low single-photon purities and unstable emission. Here we demonstrate spectrally stable, pure and narrow-linewidth single-photon emission from InP/ZnSe/ZnS colloidal QDs. Using photon correlation Fourier spectroscopy, we observe single-dot linewidths as narrow as ~5 µeV at 4 K, giving a lower-bounded optical coherence time, T2, of ~250 ps. These dots exhibit minimal spectral diffusion on timescales of microseconds to minutes, and narrow linewidths are maintained on timescales up to 50 ms, orders of magnitude longer than other colloidal systems. Moreover, these InP/ZnSe/ZnS dots have single-photon purities g(2)(τ = 0) of 0.077-0.086 in the absence of spectral filtering. This work demonstrates the potential of heavy-metal-free InP-based QDs as spectrally stable sources of single photons.