We introduce and study a new Ma–Minda subclass of starlike functions $${\mathcal {S}}^*_{\varrho },$$ defined as $$\begin{aligned} {\mathcal {S}}^{*}_{\varrho }:=\left\{ f\in {\mathcal {A}}:\frac{zf'(z)}{f(z)} \prec \cosh \sqrt{z}=:\varrho (z), z\in {\mathbb {D}} \right\} , \end{aligned}$$ associated with an analytic univalent function $$\cosh \sqrt{z},$$ where we choose the branch of the square root function so that $$\cosh \sqrt{z}=1+z/2!+z^{2}/{4!}+\cdots .$$ We establish certain inclusion relations for $${\mathcal {S}}^{*}_{\varrho }$$ and deduce sharp $${\mathcal {S}}^{*}_{\varrho }$$ -radii for certain subclasses of analytic functions.