It is known that if a forcing notion is proper, then every P-filter will generate a P-filter in the generic extension (see, e.g., [Shelah, Proper and Improper Forcing, VI.5](https://projecteuclid.org/ebooks/perspectives-in-logic/Proper-and-Improper-Forcing/toc/pl/1235419814))


On the other hand, if we start collapsing cardinals, we can destroy the P-filter property. For example, making a base of a P-point countable will add a sequence of elements from the filter (namely, a complete enumeration of that base) that serves as a counterexample for the P-filter property in the extension.


So my question is:



> 
> Are there examples of "nicer" (e.g., not collapsing cardinals) forcing notions that destroy P-filters in this way, i.e., add a sequence in the filter that the filter cannot decide?
> 
> 
> More spectacularly, is there maybe a forcing notion that could preserve a P-point as an ultrafilter while destroying the P-filter property?
> 
> 
> 


EDIT: As Martin Goldstern pointed out, I should add that I'm interested in filters on $\omega$.



Destroying the P-filter-property
(A very partial answer.) Assuming CH, the answer seems to be "no" (for filters on countable sets).


CH implies that every P-filter on $\omega$ is generated by a tower (i.e., an almost decreasing sequence) of length $\omega\_1$. In any $\omega\_1$-preserving forcing extension any countable sequence of filter sets can be refined to a countable sequence of sets from the tower, so there is a lower bound in the tower. 


