Brief Answer:
Read below.

Detailed Answer:
Hello! I am a neurologist dealing mainly with stroke and I will try to answer your question.

A stroke can be of two kinds ischemic or hemorrhagic. An ischemic stroke means a blood vessel getting blocked and the area of the nervous system it supplies as a result of not receiving vital blood is damaged, some cells are irreparably damaged, they die.
A hemorrhagic stroke means that a blood vessel ruptures and as a result there is bleeding. Due to the nervous system being confined by the skull and the vertebrae this blood can compress and destruct nervous tissue as well, leading to irreparable damage of some nerve cells and fibers.

A stroke can involve either the brain or very rarely the spinal cord blood vessels. So a spinal stroke is a stroke involving the vessels supplying the spinal cord and as a result there is damage to the spinal cord nervous pathways and cells. Symptoms and consequences may vary according to what level of the cord is involved (spinal cord extends from the neck to the lumbar region), the site and the degree.

I remain at your disposal for other questions.

---

Brief Answer:
Yes it can.

Detailed Answer:
It seems your question is incomplete as it finishes with "and". Anyway I think I understand what you mean.

Yes a spinal stroke can cause paralysis. What is involved depends on what part. If damage is high up in the cervical (neck) part it can involve the arm as well as the lower limb. If below the neck, in the thoracic or lumbar region, it doesn't involve the upper limb as the nerves towards the upper limb have already exited the spinal cord towards their final destination.
If the damage involves fibers on both sides of the cord width (depending on which vessel is involved) limbs on both sides of the body are affected. If damage affects only the right or left spinal cord symptoms are present in the limbs on that side.

I hope to have been of help.

---