The answer is **yes**.

This construction is from the paper "Pairs of Hamiltonian circuits in 5-connected planar graphs" by Joseph Zaks. This is the "connected sum" of a 5-regular planar graph with an icosahedral graph.

[![graph substitution][1]][1]

Let $G$ be a 5-valent 5-connected planar graph and $v$ a vertex of it. Replace $v$ by the 11-vertex graph shown above. The new graph is 5-valent, 5-connected, planar and has 10 more vertices. Thus there is an infinitude of such graphs.

 [1]: https://i.sstatic.net/Tf9Uv.png