The three-dimensional stability of an inflection-free boundary layer flow of length scale and maximum velocity in a stably stratified and compressible fluid of constant Brunt–Väisälä frequency , sound speed and stratification length is examined in an inviscid framework. The shear plane of the boundary layer is assumed to be inclined at an angle with respect to the vertical direction of stratification. The stability analysis is performed using both numerical and theoretical methods for all the values of and Froude number . When non-Boussinesq and compressible effects are negligible ( and ), the boundary layer flow is found to be unstable for any as soon as . Compressible and non-Boussinesq effects are considered in the strongly stratified limit: they are shown to have no influence on the stability properties of an inclined boundary layer (when ). In this limit, the instability is associated with the emission of internal-acoustic waves.