The metric tensor $g_{ij}$ of the Poincaré ball model is
$$ g_{ij} = \frac{\delta_{ij}}{(1 - x_k x^k)^2} $$
where $\delta_{ij}$ is the Kronecker delta and $x^k$ are the ambient Cartesian coordinates.
Hence the partial derivative of the metric tensor with respect to a coordinate $x^l$ is
$$ \partial_l g_{ij} = \partial_l \frac{\delta_{ij}}{(1 - x_k x^k)^2} = \delta_{ij} \partial_l (1 - x_k x^k)^{-2} = -2 \delta_{ij} (1 - x_k x^k)^{-3} \partial_l (1 - x_k x^k) = 2 \delta_{ij} (1 - x_k x^k)^{-3} (x_k \partial_l x^k + x^k \partial_l x_k) = 2 \delta_{ij} (1 - x_k x^k)^{-3} (x_k \delta_l^k + x^k \delta_{lk}) $$
In summary
$$ \partial_l g_{ij} = 2 \delta_{ij} (1 - x_k x^k)^{-3} (x_k \delta_l^k + x^k \delta_{lk}) = 4 \delta_{ij} (1 - x_k x^k)^{-3} x_l $$
The Christoffel symbols are defined in terms of the partial derivatives of the metric tensor as
$$ \Gamma^i_{jk} = \frac{1}{2} g^{il} (\partial_j g_{lk} + \partial_k g_{lj} - \partial_l g_{jk}) $$
Hence we substitute our expression for $ \partial_l g_{ij} $ with the right indices.
Is this correct? I have not been able to find an online source to verify that these are the correct Christoffel symbols for the Poincaré ball model.