  We construct and investigate Specht modules $\mathcal{S}^\lambda$ for
cyclotomic quiver Hecke algebras in type $C^{(1)}_\ell$ and $C_\infty$, which
are labelled by multipartitions $\lambda$. It is shown that in type $C_\infty$,
the Specht module $\mathcal{S}^\lambda$ has a homogeneous basis indexed by
standard tableaux of shape $\lambda$, which yields a graded character formula
and good properties with the exact functors $E_i^\Lambda$ and $F_i^\Lambda$.
For type $C^{(1)}_\ell$, we propose a conjecture.
