This is a proposed design for a solar-thermal zone refining cell. The first few would probably be delivered but the rest should be assembled using local resources.
The body of the cell is made of magnesium oxide (magnesia). This is a fairly strong material with a tensile strength between 83 and 166 MPa, compressive strength of 830 to 1660 MPa and a melting point of 3125 K (2852 °C). It has the odd property of being transparent to infrared A and B bands (0.7 to 3 micrometers), so a substantial portion of solar energy passes right through it. One reference lists about 55% of solar energy at earth's surface is infrared; in space that ratio is likely to be higher due to the lack of water absorption. The A and B bands are a small portion of the infrared spectrum and I don't have a value for the energy fraction in this range, but the specific amounts are not important at this stage.
The device could theoretically melt every element but tantalum, osmium, rhenium or tungsten (all over 3290 K). Finding a lot of those metals in your ore is a problem worth having. In practice the supporting equipment probably won't be able to handle anything much over perhaps 2200 K. Fortunately that doesn't rule out very many materials; molybdenum, niobium and rhodium are the main ones. This temp is just high enough for chromium, vanadium and platinum. Very nearly all useful materials are still within reach. One concern is that some elements vaporize before others melt; it may be necessary to do a high-temperature purification step in another device (or in this device with a means of extracting the gases) before proceeding to zone refining at very high temperatures.
A zone refining device is efficient when the melt zones are small, close together and travel quickly along the charge. It is effective when the charge is very long compared to its thickness and when at least 20 zone passes occur. It is power-efficient when the melt zones are at exactly the liquidus temperature and the solid zones are at exactly the solidus temperature, but for operational reasons this requires a few degrees of swing. The goal is to eliminate as much heat loss as possible. There is a certain minimum energy required, which is because we need to deliver the heat of formation to make the material melt and then remove that same heat to make it solidify. There are practical limits on heat retention simply because the material we want to melt is in direct contact with the material we want to keep solid.
The choice of magnesia as a wall material introduces an important method of heat loss: radiation in the upper infrared directly through the wall of the device. The solution is contained in the problem: infrared radiation can be reflected back into the charge without heating the outside of the wall.
The heat source for this device is a large solar reflector. Since the device should be long and thin a parabolic trough may be the most efficient form. However, we don't want to heat the entire length evenly; we want to heat alternating sections. That could be done using mobile reflectors between the trough and the device or by making trough sections that can change their focal point along the length of the device. It could also be done with a fixed reflector geometry that makes hotspots and then moving the entire reflector or the entire device. Since this is a batch process requiring a certain number of zone passes, let's assume the reflector is on a rail or is otherwise mobile. The device is loaded with material to refine, then heated to just below melting in a flat spot in the reflector. The active part of the reflector is a series of flat-bottomed troughs; the cylindrical flat section reflects any radiated heat from the heating zone back into itself, while the parabolic trough walls are angled to concentrate sunlight onto the heating zone. The cooling zone is above the angled parts so no sunlight is reflected into the zone. The reflector is then moved on a rack or rail or cable system until the required number of zones have passed.
The heat sink for this device could be passive radiation. That would require modeling to make certain it is feasible, but I'd bet it could be made to work. Another, faster option would be coolant channels built into the wall of the device, with valves to control which zones are being actively cooled. This poses a challenge: what can be used as a fluid coolant at 2200 °C? Water, CO and CO2 all dissociate. Hydrogen would attack the magnesia, liberating oxygen and leaving magnesium metal in the coolant channels. Helium seems to be the only viable option. Since the system can be closed or sealed, there would not normally be any significant gas leaks; this is important because helium is exceedingly rare away from Earth. The trouble with helium is that it will migrate through the tiniest of pores. I'm not convinced a ceramic material can reliably hold high-pressure high-temperature helium, particularly in a device built in the field. Perhaps sodium or calcium vapor could be used. For now I'll assume passive radiation with a possible future model using active cooling.
Performance numbers will require modeling. I'm not sure how I will get that done with no resources, but it's something to consider another day.
The output of the device is a long bar of material, separated into pure elements in order of their melting points. More or less. I've had trouble finding conclusive statements that no eutectic mixtures are encountered, so if anyone has a definite reference I would appreciate it. Further processing is necessary since we do not necessarily know the exact composition of the starting material. Some kind of elemental analysis (neutron or x-ray spectrometer, mass spectrometer, etc.) is used to identify boundaries between materials; these boundaries are cut with enough margin on either side to give pure bars of metal and several impure slices from boundary layers. These slices can be added to the next charge or stockpiled for later use; rare elements will tend to accumulate in the boundary slices and at the ends of the cylinder, so over several cycles of refining these rare materials can be concentrated until there are useful quantities available.
Bulk slices of refinery bars represent very pure materials, including semiconductor-grade silicon. These can be used to make strong alloys with precise chemical formulas and can also be used to make semiconductor devices like light-emitting diodes and photovoltaic cells. This is a reagent-free approach to pure inputs for many processes using what is available in space: dry dirt and sunlight.