  We study snarks whose edges cannot be covered by fewer than five perfect
matchings. Esperet and Mazzuoccolo found an infinite family of such snarks,
generalising an example provided by Hagglund. We construct another infinite
family, arising from a generalisation in a different direction. The proof that
this family has the requested property is computer-assisted. In addition, we
prove that the snarks from this family (we call them treelike snarks) have
circular flow number five and admit a 5-cycle double cover.
