  At room temperature, bulk La$_{7/8}$Sr$_{1/8}$MnO$_3$ is in the dynamic
Jahn--Teller O$^*$ phase, but undergoes a transition to a static, magnetically
ordered Jahn--Teller phase at lower temperatures. Here we study a $6$ unit
cells thin film of this compound grown on SrTiO$_3$, resulting in small
compressive strain due to a lattice mismatch of $\lesssim 0.2\%$. We combine
X-ray absorption spectroscopy with multiplet ligand field theory to study the
local electronic and magnetic properties of Mn in the film. We determine the Mn
$d_{3z^2-r^2}$ orbital to be $0.13\;\text{eV}$ lower in energy than the
$d_{x^2-y^2}$, which is a disproportionately large splitting given the small
degree of compressive strain. We interpret this as resulting from the strain
providing a preferential orientation for the MnO$_6$ octahedra, which are
strongly susceptible to such a deformation in the vicinity of the phase
transition. Hence, they collectively elongate along the $c$ axis into a static
Jahn--Teller arrangement. Furthermore, we demonstrate the strongly covalent
character of La$_{7/8}$Sr$_{1/8}$MnO$_3$, with a contribution of nearly $50\%$
of the one-ligand-hole configuration $d^{5} \underline{L}^1$ to the ground
state wavefunction. Finally, we find the system to be in a high-spin
configuration, with the projection of the local magnetic moment on the
quantization axis being about $3.7\;\mu_{\text{B}}/\text{Mn}$. We show,
however, that the system is close to a high-spin--low-spin transition, which
might be triggered by crystal field effects.

Strongly Correlated Electrons;Materials Science