Galois representation of an elliptic curve over a function field of char p.

Question

Let $E/K$ be a non-isotrivial elliptic curve over a function field of characteristic $p$. Let $$\rho_{E,l}:Gal(K^{sep}/K)\to GL_2(\mathbf{Z}_l)$$ For $l\neq p$, the image of $\rho_{E,l}$ is a $l$-adic Lie group closed in $GL_2(\mathbf{Z}_l)$ (see https://arxiv.org/abs/0804.1425). What is the situation for $l=p$?