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We consider the spectrum of a $U(1)$ quantum link model where gauge fields are realized as $S=1/2$ spins and demonstrate a new mechanism for generating quantum many-body scars (high-energy eigenstates that violate the eigenstate thermalization hypothesis) in a constrained Hilbert space. Many-body dynamics with local constraints has attracted much attention due to the recent discovery of non-ergodic behavior in quantum simulators based on Rydberg atoms. Lattice gauge theories provide natural examples of constrained systems since physical states must be gauge-invariant. In our case, the Hamiltonian $H={\cal O}_{\rm kin}+λ{\cal O}_{\rm pot}$, where ${\cal O}_{\rm pot}$ (${\cal O}_{\rm kin}$) is diagonal (off-diagonal) in the electric flux basis, contains exact mid-spectrum zero modes at $λ=0$ whose number grows exponentially with system size. This massive degeneracy is lifted at any non-zero $λ$ but some special linear combinations that simultaneously diagonalize ${\cal O}_{\rm kin}$ and ${\cal O}_{\rm pot}$ survive as quantum many-body scars, suggesting an ``order-by-disorder'' mechanism in the Hilbert space. We give evidence for such scars and show their dynamical consequences on two-leg ladders with up to $56$ spins, which may be tested using available proposals of quantum simulators. Results on wider ladders point towards their presence in two dimensions as well.