Let k be a non perfect field. In this article, we study the Picard group of the smooth connected unipotent k-algebraic groups. A smooth connected unipotent k-algebraic group is a form of the affine n-space A n k. To study the Picard group of a form X of A n k with a geometric method, we define a restricted Picard functor. We show that, if X admits a regular completion, then the restricted Picard functor is representable by a smooth unipotent k-algebraic group. Then, we generalise a result of B. Totaro: if k is separably closed and if the Picard group of a smooth connected commutative unipotent k-algebraic group is nontrivial then it admits a nontrivial extension by the multiplicative group. Moreover, we show that the Picard group of a unirational form of A n k is finite.