It is well known that vanilla fixed for floating swaps usually have a bit of gamma, but does a floating for floating (basis) swap have any? For the sake of simplicity, let's assume that both legs of the swaps are in the same currency.
Yes, if the two rates belong to two different currencies having different yield curves. Or in fact any two indices (bases) having different yield curves. e.g OIS vs LIBOR or LIBOR vs UST etc.
In practice a 3s-1s basis swap has negligible gamma. Imagine putting on the basis swap, then the basis swap market moves. The resulting profit or loss is the present value of a fixed annuity, whose value depends on outright rates, not basis swaps. Sometimes there could be a correlation between rates and basis swaps, which could make this covariance feel like a gamma position, but this is relatively rare in my experience.