The zero-error capacity of state-dependent channels with noiseless feedback is determined, under the assumption that the transmitter and the receiver are allowed to use variable-length coding schemes. Various cases are analyzed, with the employed coding schemes having either bounded or unbounded codeword lengths and with state information revealed to the encoder and/or decoder in a strictly causal, causal, or non-causal manner. In each of these settings, necessary and sufficient conditions for positivity of the zero-error  capacity are obtained and it is shown that, whenever the zero-error capacity is positive, it equals the conventional vanishing-error capacity. A comparison of the results with the recently solved fixed-length case is given.