No, we do not have an immortal soul. When people die they cease to exist. Dead people are not able to think, or do anything, or feel anything. They are unconscious, as if in a deep dreamless sleep. (See Psalm 6:5; Psalm 49:12, 14, 20; Psalm 146:3-4; Ecclesiastes 9:5-6;John 11:11-14).
The Bible does not mention an "immortal soul". It uses the word "soul" for anything alive — man, animal, bird, fish, insect. The Bible teaches that souls can die and will die.
"Living creature" in Genesis 1:21 and 1:24 are the same words as "Living soul" in Genesis 2:7. The same word is used again in Numbers 6:6 where "dead person" is "dead soul" in the original text. (See Numbers 31:28; Ezekiel 18:4, 20; Psalm 89:48).
Is there any hope at all for us beyond this present life? Yes, but only in Jesus Christ. He has overcome both sin and death. When he comes again to the earth, the dead will be raised and judged. Those who have been faithful to him will be blessed with eternal life in his Kingdom. (See Timothy 1:10; 1 Corinthians 15:21-23; Daniel 12:2-3; John 5:28-29; Matthew 25:31-34).
Here is a link to information which will be of assistance to you: After Death, What?
I hope you have found this helpful.