Yes - this statement is true for any irreducible representation of a split reductive p-adic group $G$, where $U$ is the unipotent radical of the Borel (a Borel subgroup is a minimal algebraic subgroup $B\subset G$ such that $G/B$ is proper; $G$ is split if such a $B$ is defined over your base field). An example is the preimage of $U$ for the standard embedding of $O(n, F), Sp(n, F),$ etc. in $GL(n).$ The source I know this from is <a href="http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/Bernst_Lecture_p-adic_repr.pdf"> Karl Rumelhart's transcription of Bernstein's Harvard course</a> (although the notes only include full proofs for $GL_n$). The general sequence of arguments this follows from is as follows: <ol><li>an irreducible representation $V$ of a $p$-adic group $G$ is <i>admissible</i> (has finite-dimensional subspace $V^K$ of vectors fixed by any open compact $K\subset G$, and is generated over $G$ by $V^K$ for $K$ sufficiently fine).</li>
<li>If $P$ is a parabolic subgroup of $G$ (such that $G/P$ is compact), $U\subset P$ is its unipotent radical and $L=P/U$ (always a reductive group, called the <i>Levi</i>) then the Jacquet restriction $V_{U,\theta}$ is admissible as a representation of $L$.</li>
<li>An admissible representation of a torus $T$ (i.e., $T(F)$ for $T$ a connected reductive commutative algebraic group) is finite-dimensional. Now just take $T=B/U$.</li></ol>