Let $E$ be an elliptic curve defined over a number field $F$. The paper concerns the structure of the $p^{\backslash}\infty$-Selmer group of $E$ over $p$-adic Lie extensions $F_{\backslash}\infty$ of $F$ which are obtained by adjoining to $F$ the $p$-division points of an abelian variety $A$ defined over $F$. The main focus of the paper is the calculation of the Gal$(F_{\backslash}\infty/F)$-Euler characteristic of the $p^{\backslash}\infty$-Selmer group of $E$. The main theory is illustrated with the example of an elliptic curve of conductor 294.