In the product of the closet unit interval $I$ with the Cantor set $C$, identify $(0,x)$ and $\big(1,f(x)\big)$ where $f(x):=3x \mod 1$.
The resulting space $X$ is the mapping cylinder of the continuous map $f:C\to C$. It is a compact metric space locally homeomorphic to $]0,1[\times C\\ ,$ thus not locally connected at any point. End-points of $I$ have not a special role; we may equivalently obtain $X$ with a larger quotient, $(\mathbb{R}\times C) / \{ (t,x)=\big (t+1,f(x)\big) \} $; the map . The important feature of the map $f:C\to C$ is that it has a dense orbit $f^n(x_0)$. This is easily seen as it is conjugate to the left-shift map on binary strings on the space ${\bf 2}^\mathbb{N}$, $(c_1,c_2,\dots)\mapsto(c_2,c_3,\dots)$ which is just how we see $f$ on the 2-digits representations of points of the Cantor set. As a consequence, the image of $\mathbb{R}\times \{x\}$ in the latter quotient is a path-connected dense subset of $X$, which is therefore connected.