Poonen proved an analogue for Drinfeld modules of the Mordell-Weil theorem. We shall generalize his results to the case of t t -modules which have the height property. Moreover, we also prove a Mordell–Weil theorem for certain t t -modules which lack the height property. In particular, let A A be the polynomial ring in one variable θ \theta over the finite field F q \mathbb {F}_q with the quotient field K K . Let ( E , ϕ ) (E, \phi ) be an iterated extension of the tensor powers of the Carlitz module defined over K K . We show that the F q [ t ] \mathbb {F}_q[t] -module E ( K ) E(K) is isomorphic to the direct sum of its torsion module, which is finite, with a free F q [ t ] \mathbb {F}_q[t] -module of rank ℵ 0 \aleph _0 .