In the context of different spacetimes and not spacetime intervals within the same spacetime. The definition could be extended as following:
Simply put, time-like or light-like is our known spacetime where causality is dictated by the $c$ speed of light upper limit inside. Thus for example, light-like in the vacuum where the speed of light is $c$ and time-like inside a transparent medium like glass where the speed of light is lower than $c$.
Space-like is a hypothetical spacetime where causality is dictated by faster than light FTL speed limit and when projected in our time-like or light-like spacetime this wouldcould even resemble to "Instantaneous action at a distance" or else known by famous Einstein as "Spooky action at a distance".
However, such a spacetime domain has not yet being discovered within our cosmos and therefore not regarded today as a possible explanation for the observed Quantum Entanglement phenomena.
Nevertheless, Dark Energy seems to have a space-like spacetime behavior.