We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant $K$-group of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum $K$-theory $QK_T(G/B)$ of the flag manifold $G/B$; this has been a longstanding conjecture about the multiplicative structure of $QK_T(G/B)$. Moreover, in type $A_{n-1}$, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivariant) quantum $K$-theory $QK(SL_n/B)$; we also obtain very explicit information about the coefficients in the respective Chevalley formula.
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