A space X has the local fixed point property LFPP, (local periodic point property LPPP) if it has an open basis $\mathcal{B}$ such that, for each $B\in \mathcal{B}$, the closure $\overline{B}$ has the fixed (periodic) point property. Weaker versions wLFPP, wLPPP are also considered and examples of metric continua that distinguish all these properties are constructed. We show that for planar or one-dimensional locally connected metric continua the properties are equivalent.
