Does “locally connected and path-connected” imply locally path-connected?

Some friends and I discovered this question when we were studying for an exam and were trying to find examples for all combinations of topological properties we had seen in the course so far. One we couldn't answer was, if there is a space that has the three properties:
-it is locally connected
-it is pathwise connected
-it is NOT locally pathwise connected
Even after looking at several books and asking our professor, we could neither find an example nor prove that such a space cannot exist.
The cone on $\mathbb{R}_{cocountable}$ should work.