Amp will be limited by noise in a single range; with switched ranges (i.e., for a Howland current pump*, a selectable series output resistor), a TL072 wouldn't be bad. I'd think that full range should be reasonable with it.
*The traditional form of this is with symmetrical resistors, but that fully halves the compliance range. Fortunately, the "top" resistors can be reduced proportionally (or something like that, I forget?) to give the same result with compliance range arbitrarily close to the amp's actual output range. Note that this exaggerates errors (increased noise gain), making the output noisier, and making trimming attractive (for amp offset and resistor tolerance).
Otherwise, there are small JFETs, like PN4116 series, which would actually have trouble reaching 10mA so you might pick something not quite as tiny, but that are guaranteed for very low leakage. Small BJTs have good performance, but aren't often rated for it; for example a typical 2N4401 leaks nanoamps at room temperature, but you'll never find a datasheet that agrees. Apparently some RF types are quite good (which makes sense from the small junction size), but be careful about voltage rating (it's often low; that's part of how they get the speed up) and parasitic oscillation (probably use a ferrite bead or resistor in series with the base/gate).
MOSFETs are similar, usually pretty good but not always guaranteed as such, and smaller is better. 2N7000 is pretty much your introductory jellybean MOSFET, and typically leaks ~nA when fully cut off. (Note that subthreshold current flow applies -- there isn't a hard pinch-off for MOSFETs, the drain current just keeps falling exponentially below the quadratic current flow region. Vgs(th) is defined at some modest current in this range; expect to need Vgs(off) much lower than this. For enhancement mode MOSFETs, drain current is usually low enough to be considered leakage current, for Vgs near zero -- no negative bias needed.)
And not that you would, but just for completeness: vacuum tubes leak quite a lot, I think due to their hot operation (weak emission from everywhere, not just the intended cathode surface). The dynamic range (cutoff to full conduction) isn't very good, certainly fewer orders of magnitude than you are asking for here. Though it would be interesting, I guess, to think of ways to trick it into working anyway. Like, the heater voltage could be reduced on the low ranges, to reduce current capacity and gain, and hopefully leakage. Or maybe a high voltage or RF type gives lower leakage, again in normal or reduced operation. If very low bandwidth is acceptable, a high voltage diode could certainly be used, varying the heater voltage to set output current (the thermal time constant of such a heater is fractional seconds).
Tim