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 

<cite authors="Montgomery, Deane; Zippin, Leo">_Montgomery, Deane; Zippin, Leo_, Topological transformation groups, Mineola, NY: Dover Publications (ISBN 978-0-486-82449-9). xi, 289&nbsp;p. (2018). [ZBL1418.57024](https://zbmath.org/?q=an:1418.57024).</cite>

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.