Let X be a compact Hausdorff topological space and let C X and C R X denote the complex and real Banach algebras of all continuous complex-valued and continuous real-valued functions on X under the uniform norm on X, respectively. Recently, Fupinwong and Dhompongsa 2010 obtained a general condition for infinite dimensional unital commutative real and complex Banach algebras to fail the fixed-point property and showed that C R X and C X are examples of such algebras. At the same time Dhompongsa et al. 2011 showed that a complex C * -algebra A has the fixed-point property if and only if A is finite dimensional. In this paper we show that some complex and real unital uniformly closed subalgebras of C X do not have the fixed-point property by using the results given by them and by applying the concept of peak points for those subalgebras.