Semiconductor switching device with extremely low leakage current in off state
A possible candidate might be an analogue switch such as the ADG601/602: -



It has an on-resistance of 2.5 ohms and a leakage typically of 10 pA at 25 degC.


  current to switch: at most 1A, probably much less


Well, 1 amp would be fatal to the ADG60x because it has an on-resistance of 2.5 ohms but, you say later on this: -


  resistance in the on state: may be relatively high, even 1kOhm is
  probably good


And if 1 amp passed through 1 kohm it would dissipate 1000 watts so you need to consider what you might really need.

But you also say this: -


  voltage to switch: very low, between 0.1V and 10V


And, to be able to use an analogue switch, it requires that the voltage supply exceeds the signal levels, so maybe the ADG1201 might be more suitable for this voltage range. It can operate from much larger supply rails (+/- 15 volts or +12 volts) but has an on-resistance of typically 120 ohm.

 The ADG601/602 can operate from dual 5 volt rails but only a single positive rail of 5 volts. However, you might be able to bias it up to get a 10 volt operating range.

Food for thought.