Does current flow between a positively charged conductor and a neutral conductor?
You have a 9V battery => You have a device which generates an electric field between its terminals. 

You connect a metal wire to the +terminal => the field draws free electrons from the wire towards the +terminal until the 3 forces find a balance. They are:


the attraction from the metal atoms of the wire
the field of the battery
the random motion phenomena which is called "heat" (this can be omitted if we do not assume remarkable temperature differences)


If the wire happens to be long, the lack of electrons in the distant end is effectively a +charge which generates a field around the wire. So the field generated by the battery can be noticed and utilized as far as your wire enters without disconnecting it from the battery. This is why we generally have wires.

The electric field is around a charge, there's no need to have any other pole nor negative charge. 

Voltage is a practical measure for electric field. It's in use because batteries and other electricity sources generally cannot keep a certain electric field strength, but they can give a certain energy to the electrons that they put in motion. In batteries this is because the chemical reactions release a certain energy if the current is allowed to flow. 9V battery gives to moving electrons energy = 9 Joules per one coulomb charge or as well 9 electronvolts per one electron. To get that energy into use  you must have a current loop, because every electron that the +terminal sucks must be also removed from the battery. They come out from the -terminal and the circulation prevents charge buildup, which would stop the current as soon as the balance is found.

The strength and spatial form of the electric field that the battery generates is highly dependent on the forms and placements of those parts which are connected to the battery terminals. Battery cannot keep certain electric field, it can give only a certain energy to the electrons. The voltage is the measure for that ability.

Without a conducting current loop the current stops as soon as the balance is found.What is the balance, do we have any measures for it? Yes, we have. We can calculate the capcacitance between two conductive pieces or as well from one piece to the infinity in otherwise empty space. The capacitance tells how much charge a certain voltage can place to the conductive piece which is not connected to both terminals of a battery. 

If you happen to have two separated metallic pieces and you connect +terminal of the 9V battery to piece A and the  -terminal to piece B, the current flows until the balance is found. Let's assume the 9V battery can push 9 coulombs (exactly -9C) of electrons to B and respectively pulls as much out of A until all forces are in balance. What do you have?

Congratulations! you have one farad capacitor and it's charged to 9V. It's not especially little.