The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar $δ$-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed $δ\geq 0$, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set $Z$) with any subset of the connected components of $G - Z$ induces again a planar $δ$-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is $\mathrm{poly}(δ) \cdot \log n$. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar $δ$-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant $δ$, running in $n\, \mathrm{polylog}(n) \cdot 2^{\mathcal{O}(δ^2)} \cdot \varepsilon^{-\mathcal{O}(δ)}$ time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no $n^{o(δ)}$-time algorithm on planar $δ$-hyperbolic graphs, unless ETH fails.