They answer is yes and no (depending on what you mean by stable).
John Rennie has given a good intuitive picture of why orbits inside the horizon could be possible is the black hole is rotating or spinning. Let me give a slightly more technical answer. For this lets focus on the astrophysically relevant case of a rotating black hole. In general relativity the metric of a (perfect) rotating black hole is given by the Kerr metric.
The radial motion of a particle in the Kerr metric is controlled by the radial potential:
$$ R = (E(r^2+a^2)-a L)^2 - (r^2-2r+a^2)(r^2+(aE-L)^2+Q),$$
where $E$ is the (specific) energy, $L$ the (specific) axial angular mometum, and $Q$ is the so-called Carter constant. (The mass of the black hole has been normalized to $M=1$) Geodesics oscillate between the zeros of this polynomial (in regions where R is negative). In general, since R is a fourth order polynomial in $r$ it has four roots, and potentially two oscillating solutions for each choice of $a$, $E$, $L$, and $Q$ (for some choices there may be only 2 or even no zeroes with real values of $r$, in which case we may have 1 or no oscillating solutions).
It is fairly trivial to confirm that choices of $a$, $E$, $L$, and $Q$ exist such that this polynomial has zeroes inside the inner horizon. For example, for $a=0.9$, $E=0.1$, $L=0.1$, and $Q=0$, $R$ has roots
\begin{align}
r_1 &= 1.4577\\
r_2 &= 0.562259\\
r_3 &= 0.000246485\\
r_4 &= 0\\
\end{align}
and R is negative between $r_2$, and $r_3$. This means that there exists a timelike geodesic that oscillates between $r_2$, and $r_3$, both of which lie inside the inner horizon $r_{-} = 0.56411$.
Such orbits are stable in the sense that any small perturbation of the initial conditions (and thus $E$, $L$, and $Q$) would again lead to an oscillating solution. (This follows from the fact that the positions of the roots are smooth functions of $E$, $L$, and $Q$).
However, stability of the geodesic solution is only part of the stability question. While the Kerr metric outside the event horizon is thought to be stable to gravitational perturbations, this is known to be false for the part of the Kerr metric inside the (outer) event horizon. In particular, the area around the inner horizon is highly unstable with linear perturbations tending to blow up and becoming singular at the inner horizon. Consequently, the part of the Kerr metric inside the inner horizon should not be taken seriously from an astrophysical point of view, as any perturbation (such as that caused by the presence of planet in the interior) could lead to a wildly different metric.
So while these orbits maybe stable as geodesics in a fixed metric, the metric itself is unstable to perturbations. Consequently, we cannot consider these solutions as realistic "stable orbits", and any speculation of civilizations existing on a planet on such an orbit is the stuff of fairy tales.
