Consider a symplectic manifold $(M,\om)$, a closed coisotropic submanifold $N$ of $M$, and a Hamiltonian diffeomorphism $\phi$ on $M$. A leafwise fixed point for $\phi$ is a point $x\in N$ that under $\phi$ is mapped to its isotropic leaf. These points generalize fixed points and Lagrangian intersection points. The main result of this talk will be that $\phi$ has a leafwise fixed point, provided that it is the time-1-map of a Hamiltonian flow whose restriction to $N$ stays $C^0$-close to the inclusion $N\to M$. The method of proof of this result leads to a local coisotropic version of Floer homology.