  We consider ground states solutions $u \in H^2(\mathbb{R}^N)$ of biharmonic
(fourth-order) nonlinear Schr\"odinger equations of the form $$ \Delta^2 u + 2a
\Delta u + b u - |u|^{p-2} u = 0 \quad \mbox{in $\mathbb{R}^N$} $$ with
positive constants $a, b > 0$ and exponents $2 < p < 2^*$, where $2^* =
\frac{2N}{N-4}$ if $N > 4$ and $2^* = \infty$ if $N \leq 4$. By exploiting a
connection to the adjoint Stein--Tomas inequality on the unit sphere and by
using trial functions due to Knapp, we prove a general symmetry breaking result
by showing that all ground states $u\in H^2(\mathbb{R}^N)$ in dimension $N \geq
2$ fail to be radially symmetric for all exponents $2 < p < \frac{2N+2}{N-1}$
in a suitable regime of $a,b>0$.
  As applications of our main result, we also prove symmetry breaking for a
minimization problem with constrained $L^2$-mass and for a related problem on
the unit ball in $\mathbb{R}^N$ subject to Dirichlet boundary conditions.
