Kernel of the local-global map for first galois cohomology of $m$-torsion group

Question

Let $E$ be an elliptic curve over a number field $K$. Are there examples when the local-global map $$H^1(K,E[m])\to \prod_\nu H^1(K_\nu,E_{K_\nu}[m])$$ has a nontrivial kernel?

I know that if we compose this map with $\prod_\nu H^1(K_\nu,E_{K_\nu}[m])\to \prod_\nu H^1(K_\nu,E_{K_\nu})[m]$ the kernel of the composition is called Selmer group and it often turnes out to be nontrivial.