A simple example is obtained by taking $P$ to mean "has positive dimension".  
Every integral local ring of positive dimension $(A,\mathfrak m)$ has $P$ at all  maximal ideals (i.e. just  at $ \mathfrak m$ !) since $A_{\mathfrak m}=A$ , but $P$ fails at the generic point $\eta=(0)$ since $A_\eta$ has dimension zero, being a field.