This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number Ro, Froude number Fh, Reynolds number Re and aspect ratio α. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number |Ro| is large and is stabilized for small and moderate |Ro| when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber m=1 when the Burger number Bu=α2Ro2/(4F2h) is larger than unity. When Bu≲0.7Ro+0.1, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for m=2 exists when Fh/α is below a threshold depending on Ro. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small Bu. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on Fh/α and Ro. Baroclinic instability develops when Fh/α|1+1/Ro|≳1.46 in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.