We study kernelization (a kind of efficient preprocessing) for NP-hard problems on planar graphs. Our main result is a kernel of size at most 9k vertices for the Planar Maximum Nonseparating Independent Set problem. A direct consequence of this result is that Planar Connected Vertex Cover has no kernel with at most (9/8 - epsilon)k vertices, for any epsilon > 0, assuming P \ne NP. We also show a very simple 5k-vertices kernel for Planar Max Leaf, which results in a lower bound of (5/4 - epsilon)k vertices for the kernel of Planar Connected Dominating Set (also under P \ne NP). As a by-product we show a few extremal graph theory results which might be of independent interest. We prove that graphs that contain no separator consisting of only degree two vertices contain (a) a spanning tree with at least n/4 leaves and (b) a nonseparating independent set of size at least n/9 (also, equivalently, a connected vertex cover of size at most 8/9n). The result (a) is a generalization of a theorem of Kleitman and West [SIDMA 1991] who showed the same bound for graphs of minimum degree three. Finally we show that every n-vertex outerplanar graph contains an independent set I and a collection of vertex-disjoint cycles C such that 9|I| >= 4n-3|C|.