As discussed in the comments to Gerry Myerson's answer, I am assuming you want to work on a sphere, so that there is a vertex assigned to the outer face. If not, then Gerry's answer gives a simple counter-example.
The statement is still false if you include the extrerior face. You want the planar dual of the Tutte graph. As discussed at the link, Tait conjectured that any three-regular three-connected planar graph is Hamiltonian. Proving this for planar graphs would have implied the 4 color theorem, so people were pretty excited about this conjecture until Tutte broke it. The smallest known counterexample has 38 vertices.
It is true that almost all three-regular graphs are Hamiltonian. Here is a survey on Hamiltonian cycles in three-regular graphs.