The already mentioned MCP1703 is about as good a fit as you could hope for.
Part number you want is MCP1703T-4002... where
- 40 indicates 4 Volts and next
- .. 0 indicates fixed voltage.
- ... 2 = 2% = std availability)
BUT best of all it's in stock at Digikey for $US0.77 in ones ($0.47/100) as MCP1703T-4002E/DBCT-ND
- Vinmax = 16V <- good for Microchip parts of this sort.
- Iout max = 250 mA for the 4V version.
- Lower Vout versions are available with lower Iout ratings.
There are 5 or more other 4V LDOs available even from Digikey but most have a zero load quiescent current in the 70 to 200 uA range.
MCP1703 Ground current (quiescent current) is 5uA max at zero load (2 uA typical). Worst case should always be used for design, 5uA is equal to your specified minimum load current, but would be hard to improve in a "roll your own" alternative on without vast effort. In particular, the availability of a 2% bandgap reference at only 5 uA quiescent current (max) places it in very select company.
For non zero loads the ground current is about 0.5 uA per ma, or a "gain" of about 2000, which is amazingly good. Overall this appears to be a superb part and good value for money as well.
Also available as std in 0.1V Vout increments from 1.2V to 5.5V and in 50 mV increments to special order.
Note that the various parameters highlighted at the top of the data sheet and on the product selection page are (as is usually the case) the best case value in each case and you usually don't get all at the same time. eg Iout max is only for >= 4V versions.
For completeness - may meet some other need better, other 4V LDOs from Digikey, none as good for this role, include:
TI (NatSemi) LP2985
TC1014 Microchip (far higher Iq)