Is a delta-hedged long call a free lunch

You hold a call and are short delta shares against it. Because the position is long gamma, a sharp move in the stock in either direction produces a profit. Does this mean you have an arbitrage?

No. The long-gamma position pays for its convexity through negative theta: if the stock barely moves, time decay erodes the option faster than rebalancing earns. In Black-Scholes, $\Theta+\tfrac12\sigma^2S^2\Gamma=r(V-\Delta S)$, so the hedged book earns the risk-free rate only when realised volatility equals the implied volatility you paid; profit requires realised to exceed implied.