This paper investigates a peculiar case of thermal convection in a vertical porous prism with impermeable and partially conducting walls. We facilitate the analysis in the numerical finite-element environment alongside with analytical considerations, in special cases where direct solutions are feasible. The present eigenvalue problem results in a non-normal-mode behaviour in the horizontal cross-sectional plane. Further, it is identified that the stagnation points for the horizontal flow are displaced from the extremal points of the temperature perturbation, for both symmetric and antisymmetric eigenfunctions. In addition, the corresponding normal-mode counterparts are provided from an analogy solution. We show that the critical Rayleigh number decreases with increasing Robin parameter values for all of the investigated aspect ratios. Finally, the influence of the aspect ratio on the critical Rayleigh number for the fully conducting wall case is identified. An asymptotic benchmark case of the Robin condition is validated from well-known analytical solutions which confirm the effectiveness of the predictions made in this paper. In fact, this is the first contribution that reports a three-dimensional geometry with a two-dimensional non-normal mode.