Let $p$ be a prime number and $ς$ and $m$ be a positive integers. Let $\mathcal{R} = \mathbb{F}_{2^m} + u\mathbb{F}_{2^m} + u^2\mathbb{F}_{2^m}$ ($u^3 = 0$). Cyclic codes of length $2^ς$ over $\mathcal{R}$ are precisely the ideals of the local ring $\frac{\mathcal{R}[x]}{\langle x^{2^ς}-1 \rangle}$. The Gray map from a code of Lee weight over $\mathbb{Z}_4$ to a code with Hamming weight over $\mathbb{F}_2$ is known to preserve weight. In this paper, we determine the Lee distance of cyclic codes of length $2^ς$ over $\mathcal{R}$.