ArcGIS fills sinks to the surrounding level and the z_limit won't help you because it effectively does the opposite of what you want. I don't recall any setting for what you need but I can think of a possible work around:
- Fill all sinks
- Subtract the original DEM from the filled DEM (makes 'Sinks Only')
- Reclassify 'Sinks Only' so that all values less than 5 are set to 0 and all values greater than 5 are set to 5
- Add the reclassified 'Sinks Only' to the original DEM
The Sink tool is another option that might help (find all sinks, reclassify to 5 and then add to original DEM) but the danger of this approach is that you might 'over-fill' a sink and create a peak.
The above method had assumed that the OP wanted the sinks that couldn't take 5m worth of water left alone. However, if you want to fill the sinks which are less than 5m deep to their pour point (i.e. without creating a peak), then don't reclassify the sinks less than 5m to zero - leave them as they are and just reclassify everything over 5m to 5m.