In general, this will be false. Examples are found among solenoidal manifolds, defined by Sullivan. For example, 1-dimensional solenoids.

Many of these are obtained by taking the inverse limit of finite-sheeted covers of a fixed manifold. The universal such example is obtained by taking a manifold $M$ with infinite residually finite fundamental group $\pi_1(M)$, and taking its profinite completion $\widehat{\pi_1(M)}$. Let $\tilde{M}$ denote the universal cover of $M$. Then $(\tilde{M}\times \widehat{\pi_1(M)})/ \pi_1(M)$ is a solenoid, where the action is by covering translation on the left and coset action on the right where $\pi_1(M)\subset \widehat{\pi_1(M)}$.

This space is locally homeomorphic to open subsets of $\mathbb{R}^n \times \widehat{\pi_1(M)}$. For an infinite residually finite group, $\widehat{\pi_1(M)} \cong \mathcal{C}$, the Cantor set. So there is a neighborhood basis of sets which are homeomorphic $\mathbb{R}^n \times \mathcal{C}$ where $\mathcal{C}$ is totally disconnected, since $\mathcal{C}$ has a neighborhood basis of sets homeomorphic to itself.