Recall that $E$, the earring space, is the union in the plane of a countable collection of shrinking circles, all tangent to the $y$ axis at the origin. To obtain the desired counterexample, form the cone on $E$.
Edit: My example just above is very similar to that of Henrik Rüping (in the comments) and works for much the same reason (also mentioned by Henrik Rüping in the comments). Seeing the further questions asked by the original poster prompted me to give the example below.
Let $S$ be the following subset of the plane:
$$S = \{(0, 0)\} \cup \{ (1/n, 0) \mid \mbox{$n > 0$ a natural number}\}$$
Let $A = (0, 1)$ be the given point in the plane, called the apex. Let $I_n$ be the closed line segment in the plane connecting the apex $A$ to the point $(1/n, 0)$. Finally, let $I_\infty$ be the closed line segment connecting the apex to the origin $(0, 0)$. We define the cone $C(S)$ to be the set
$$C(S) = I_\infty \cup \left( \cup_{n > 0} I_n \right)$$
equipped with the subspace topology. The space $C(S)$ is sometimes called the broom space. (I think it looks a bit more like a "brush" than a broom, but anyway.)
We make two claims.
Claim 1: Straight-line homotopy gives a deformation retraction of the broom $C(S)$ to its apex $A$.
Thus $C(S)$ is contractible.
Claim 2: The broom $C(S)$ is not (homologically) locally connected at the origin $(0, 0)$.