This paper considers an arbitrarily-varying fading channel consisting of one transmitter, one receiver and an arbitrarily varying adversary. The channel is assumed to have additive Gaussian noise and fast fading of the gain from the legitimate user to the receiver. We study four variants of the problem depending on whether the transmitter and/or adversary have access to the fading gains; we assume the receiver always knows the fading gains. In two variants the adversary does not have access to the gains, we show that the capacity corresponds to the capacity of a standard point-to-point fading channel with increased noise variance. The capacity of the other two cases, in which the adversary has knowledge of the channel gains, are determined by the worst-case noise variance as a function of the channel gain subject to the jammer's power constraint; if the jammer has enough power, then it can imitate the legitimate user's channel, causing the capacity to drop to zero. We also show that having the channel gains causally or non-causally at the encoder and/or the adversary does not change the capacity, except for the case where all parties know the channel gains. In this case, if the transmitter knows the gains non-causally, while the adversary knows the gains causally, then it is possible for the legitimate users to keep a secret from the adversary. We show that in this case the capacity is always positive.