I assume removing a face means here: select a face, remove *all* its vertices and *all* their incident edges. (Otherwise I don't understand what happens to edges when one of their endpoints is removed.)

Then the answer is no. Some small counterexamples are [Graph 226][1] and [Graph 160][2] (House of Graph numbering). Pictures (courtesy of HoG) below. Removing the central face from each of them leaves a triangle and a square, respectively, so the vertex connectivity goes down to two in each case.

[![A six-vertex planar 4-connected graph][3]][3]
[![An eight-vertex planar 4-connected graph][4]][4]


  [1]: https://houseofgraphs.org/graphs/226
  [2]: https://houseofgraphs.org/graphs/160
  [3]: https://i.sstatic.net/fkQadN6tt.png
  [4]: https://i.sstatic.net/Ole3oT71t.png