For the purposes of this question, a polyhedron has triangular facets.
Convex polyhedra are rigid by Cauchy’s theorem. Steffen’s polyhedron is an example of a non-convex polyhedron that is flexible (i.e., non-rigid). However, it appears to have edges of different lengths. My question: are there equilateral flexible polyhedra or are all equilateral polyhedra rigid?
Motivation: I have a bars-and-balls magnetic construction set and I would like to build a flexible polyhedron. But all the bars I have are equal in length.
As noted in a comment, a polyhedron with congruent equilateral triangles for faces is called a "deltahedron". This reference lists "Bellows deltahedra" as flexible deltahedra that seem to meet OP's criteria. It shows two distinct examples, and also links to another (possibly semi-defunct) website with additional images and descriptions of these objects. If I'm understanding the images correctly, the simplest example has $48$ faces and a $6$-fold rotational symmetry.