  We show that on the real $2$-dimensional Banach space $\ell_1^2$ there is an
analytic function $f:B_{\ell_1^2}\rightarrow \mathbb{R}$ such that its power
series at origin has radius of uniform convergence one, but for some $a\in
B_{\ell_1^2}$ the power series centred at that point has radius of uniform
convergence strictly less than $1-\|a\|$. This result highlights a fundamental
distinction in real analytic functions (compared to complex analytic
functions), where the radius of analyticity can differ from the radius of
uniform convergence. Moreover, this example provides the first non-trivial
upper bound for the constant of analyticity.
