The problems of risk sensitive portfolio optimisation under transaction costs have taken a considerable attention in the recent literature on mathematical finance. We study the associated problems of risk sensitive utility indifference pricing for perpetual American options with fixed transaction costs in the classical model of financial market with two tradable assets. Assume that the investors trading in the market must pay transaction costs equal to a fixed fraction of the entire portfolio wealth each time they trade. The objective is to maximise the asymptotic (risk null and risk adjusted) exponential growth rates based on the expected logarithmic or power utility of the difference between the terminal portfolio wealth and a certain amount of the option payoffs. It is shown that the optimal trading policy keeps the number of shares held in the assets unchanged between the transactions. In order to determine the optimal trading times and sizes, we reduce the initial problems to the appropriate (discounted) time-inhomogeneous optimal stopping problems for a one-dimensional diffusion process representing the fraction of the portfolio wealth held by the investor in the risky asset. The optimal trading and exercise times are proved to be the first times at which the risky fraction process exits certain regions restricted by two time-dependent boundaries. Then, certain amounts of assets should be bought or sold or the options should be exercised whenever the risky fraction process hits either the lower or the upper time-dependent curve. The latter are characterised as unique solutions of the associated parabolic-type free-boundary problems for the value functions satisfying the smooth-fit conditions at the curved boundaries. The optimal asymptotic growth rates and trading sizes are specified as parameters maximising the value functions of the resulting optimal stopping problems. We illustrate these results on the examples of the perpetual American call and put as well as the asset-or-nothing options, for which we obtain the utility indifference prices as well as the optimal trading and exercise boundaries in a closed form.