If I am thinking of the correct plant, it's Bolusanthus speciousus, native to south africa. Very slow growing indeed but pretty. I have one and it's just finishing flowering. I have not heard of them suckering... not saying they couldn't ... you might have to google a bit. Perhaps though they are seedlings instead?
All I can suggest is that you spade down and around the plantlets to a generous width and depth to get the best possible amount of root. If indeed you think it is coming from the parent tree, then leave that side till last and excavate carefully to see what you have to sever.
Put it in a large enough tub, backfill with potting mix, water through, maybe use some seasol, put in a semishaded position for a while and "wait and see" & then be ready to plant out later. Or plant out in your garden immediately if you so wish but try to give it some shade-cloth protection against the heat. You may get some dieback from the transplant but if the roots and trunk are OK it will probably survive and re-shoot. If that does not succeed I may get seeds off mine, but it's a very long wait for flower that way.

Though I've never heard of that tree I agree with Alison's directions. I've only one further suggestion, after you've removed the plant I'd cut away about 50% of the leaf growth and if you still have plenty of leaves I'd cut some of the larger ones in half. All that reduces the stress on the new plant so it can concentrate on making new roots and adapting to the change.