It seems reasonable that no-arbitrage doesn't necessarily imply EMH. If we are talking pure arbitrage opportunities, like offsetting the same contract on 2 exchanges, futures cash and carry, BS options no-arbitrage, etc. Mainly, for derivative products it's very easy to do the arbitrage trade. However, this means that you are assuming the underlying product is priced at fair-value.
So in theory you could have some person who is beating the index by buying /selling products not priced at fair value. However, you can still satisfy the no-arbitrage property by correctly pricing all the derivatives on that product.
There are many more examples that could satisfy no-arbitrage and not EMH.
For what it's worth the weak EMH is a plausible claim. In this case the no-arbitrage clause would be easily satisfied.
The semi-strong EMH doesn't make sense in practice. For a market to adjust prices to new information instantaneously is impossible. Information would have to dissipate from it's source to every other market participant simultaneously. Since current technology doesn't allow for this we end up with a some participants being fast, and others being slow. The fast guys end up making their money from this inefficiency, even if it only lasts for microseconds.