Computational complexity of finding a spanning tree in planar hybergraphs
The answer is in this paper "Tree-Residue Vertex-Breaking: a new tool for proving hardness"


  The Hypergraph Spanning Tree problem in $k$-uniform $2$-regular
  hypergraphs is $NP$-complete for any $k≥4$, even when the incidence graph
  of the hypergraph is planar.


Basically, they define a new problem: 


  Tree-Residue Vertex-Breaking (TRVB): given a multigraph $G$ some of
  whose vertices are marked "breakable," is it possible to convert $G$
  into a tree via a sequence of "vertex-breaking" operations (replacing
  a degree-$k$ breakable vertex by $k$ degree-$1$ vertices, disconnecting the
  k incident edges)?


Proving this:


  For any k≥4, TRVB is NP-complete when the given multigraph is
  restricted to be planar and consists entirely of degree-k breakable
  vertices. 


You can see that solving TRVB for the incidence graph of a hypergraph corresponds to solving SPANING TREE in hypergraphs. Where the incidence graph of a hypergraph is isomorphic to the bipartite representation of the hypergraph described above.