Say that a space (= compact metrizable space) $X$ is locally strictly contractible if, for every $p\in X$ and neighborhood $U$ of $p$, there is a neighborhood $V$ of $p$ which can be contracted to $p$ within $U$ relative to $p$, i.e. by a homotopy leaving $p$ fixed. (This may be bad terminology, and I welcome alternatives.)
Every ANR is locally strictly contractible. What about the converse? Borsuk's example of a locally contractible space which is not an ANR is not locally strictly contractible.
Incidentally, "locally contractible" is defined in different ways in various references. What is the currently accepted definition?