This is an addendum to Pedro's answer:

Theorem 1. Suppose that $G$ is a locally compact (Hausdorff) topological group. Then $G$ is locally homeomorphic to the product of a totally disconnected space and ${\mathbb R}^n$.

A reference for this result (that should be better known) is

Montgomery, Deane; Zippin, Leo, Topological transformation groups, Mineola, NY: Dover Publications (ISBN 978-0-486-82449-9). xi, 289 p. (2018). ZBL1418.57024.

section 4.9.3. (The actual theorem says a bit more, namely that locally, at $e$, $G$ is a product of a local Lie group and a totally disconnected group.

In the same book you will also find Theorem 4.10.1 answering your original question:

Theorem 2. If $G$ is a locally compact, locally connected and finite dimensional topological group, then $G$ is a Lie group (possibly non-2nd countable).

Of course, this is a consequence of Theorem 1.

This is an addendum to Pedro's answer:

Theorem 1. Suppose that $G$ is a locally compact finite dimensional (Hausdorff) topological group. Then $G$ is locally homeomorphic to the product of a totally disconnected space and ${\mathbb R}^n$.

A reference for this result (that should be better known) is

Montgomery, Deane; Zippin, Leo, Topological transformation groups, Mineola, NY: Dover Publications (ISBN 978-0-486-82449-9). xi, 289 p. (2018). ZBL1418.57024.

section 4.9.3. (The actual theorem says a bit more, namely that locally, at $e$, $G$ is a product of a local Lie group and a totally disconnected group.

In the same book you will also find Theorem 4.10.1 answering your original question:

Theorem 2. If $G$ is a locally compact, locally connected and finite dimensional topological group, then $G$ is a Lie group (possibly non-2nd countable).

Of course, this is a consequence of Theorem 1.