  The identity by Chaundy and Bullard expresses $1$ as a sum of two truncated
binomial series in one variable where the truncations depend on two different
non-negative integers. We present basic and elliptic extensions of the
Chaundy--Bullard identity. The most general result, the elliptic extension,
involves, in addition to the nome $p$ and the base $q$, four independent
complex variables. Our proof uses a suitable weighted lattice path model. We
also show how three of the basic extensions can be viewed as B\'ezout
identities. Inspired by the lattice path model, we give a new elliptic
extension of the binomial theorem, taking the form of an identity for elliptic
commuting variables. We further present variants of the homogeneous form of the
identity for $q$-commuting and for elliptic commuting variables.
