Existence of extension of isometries on subspaces of $\ell_{p}(\mathbb N)$.

Question

Consider $\ell_{p}(\mathbb N)$ with $p\in\{1,\infty\}$. Let $Y\subseteq\ell_{p}(\mathbb N)$ be a closed linear subspace and $\varphi\colon Y\rightarrow Y$ a (not necesserily surjective) isometry.

Can this isometry be extended to an isometry, i.e., $\overline{\varphi}:\ell_{p}(\mathbb N)\rightarrow\ell_{p}(\mathbb N)$ with $\overline{\varphi}\mid Y=\varphi$?