Paraffin wax, perhaps?
What happens if you put a match or cigarette lighter near it?
Does it melt?
Does it catch fire?
Does it do both or just one of those?
If it does neither, how hard is it - can you scratch it with a knife?
(Almost certainly, the surface looks different to the interior because of exposure to 'the elements'.)
Just tried burning some. It is quite hard but I managed to chip a small lump off using a knife and a lot of force. It burnt readily with a yellow flame and thick, black, sooty smoke. There was no sign of it melting but the smoke was acrid and most unpleasant.
Lack of melting points away from paraffin/other wax and away from thermoplastic but somewhat towards thermosetting plastics (eg. resins), but the style of fracture is not entirely consistent with that. I'm leaning towards a hydrocarbon-related solid like napthalene (which could arise on a beach from spilled cargo), or hexamine (as used in firelighter blocks, but they are low-smoke). Further than that, I'm out of suggestions, sorry.
Somebody else may be able to recognize it as of animal origin, perhaps, as you hint.
- Lumps of wax can end-up on beaches as a result of oil-spill clean-up operations.
- It is not a nylon or other similar white solid plastic because they do not have a strong odour.
- I should have given you warning about heating/burning your sample regarding flammability and gases; my apologies.
Ambergris (from whales) may be worth considering, see following link:
Could be a rockpool encrustation, 'lithothamnia' (=lithophyllum sp's or Lithothamnium sp's) . They are pink & coat rockpools but bleach white after exposure. Collins' seashore of Britain & Europe ISBN 0002199556 (1996) P.29 shows a white encrustation very like this.