Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ? 

**Edit:** First, I would like to thank Fernando, Vladimir and Jesper for the excellent answers!
My original motivation was the following question:
Suppose that $\mathcal{A}_{\infty}$ is a cofibrant replacement of the associative operad $\mathcal{A}$ in the category $\mathsf{Op}$ of topological operads (non-symmetric) and suppose that $X$ is a finite connected $CW$-complex. Is it true that $$Hom_{\mathsf{Op}}(\mathcal{A}_{\infty},End(X))\neq \emptyset \Rightarrow Hom_{\mathsf{Op}}(\mathcal{A},End(X))\neq \emptyset ?$$

Here is a counterexample: Suppose that $M$ is a smooth closed connected manifold with a structure of $\mathcal{A}_{\infty}$-space (i.e, Loop space) such that $M$ is not homotopy equivalent to a Lie group. By definition $Hom_{\mathsf{Op}}(\mathcal{A}_{\infty},End(M))\neq \emptyset$. On another hand if $Hom_{\mathsf{Op}}(\mathcal{A},End(M))\neq \emptyset$, this will implie that $M$ has at least one structure of topological monoid. By Wallace Theorem any closed manifold with a structure of topological monoid is a Lie group, hence $M$ is a Lie group. Contradiction.