Do all planets have cores, or is it just rock planets, and if so are they solid?

Question: Do all planets have cores, or is it just rock planets, and if so are they solid?
- Keywords:
  core,
  gas,
  planet,
  rock
Asked by tala123 to Lidunka on 14 Jun 2015. This question was also asked by Millie Hollidge ;).
- Lidunka Vocadlo answered on 14 Jun 2015:
  
  all planets have cores, but what they are made of is probably different. The terrestrial planets (mercury, venus, earth, mars) have iron alloy cores. The gas giants are very large so they have such high pressures in their deep interiors that they make metallic hydrogen cores with rocky centres made of who-knows-what!

Question: Do all planets have cores, or is it just rock planets, and if so are they solid?