Hamiltonian cycle on graphs without small cycles While answering this question on cstheory, I (informally) proved on the fly the following theorem:

Theorem: For any fixed $l \geq 3$ the Hamiltonian cycle probem remains NP-complete even if restricted to planar bipartite undirected graphs of maximum degree 3 that don't contain cycles of length $\leq l$.

It seems very unlikely that it has not already appeared somewhere.
But it allows to settle many Hamiltonian cycle/path problems on graphclasses.org that are marked as "Unknown to ISGCI" (see for example this one); indeed a direct corollary is that Hamiltonian cycle and path problems are still NP-complete if restricted to $(H_1,...,H_k)\text{-free}$ graphs, where each of the $H_i$ contains at least one cycle.


  Can you give me a reference of the paper/book where it appeared?


(then I'll contact people at graphclasses.org)
This unpublished manuscript by Hougardy, Emden-Weinert and Kreuter in 1997 
provided a simple proof for the following result which is much stronger than the result pointed out in 
Kristoffer Arnsfelt Hansen's answer:


  For any given rational number $0\le r &lt;1/2$, the Hamiltonian cycle probem remains NP-complete even if restricted to bipartite planar $n$-vertex graphs of maximum degree 3 and girth $\ge n^r$.


The manuscript contains also similar results for other problems such as Dominating set, Max cut, VFS, etc.