The text appears to indicate that Yona physically died at the point that the great fish swallowed him.
Yona indicates that he cried from the depths of "Sheol" (Jonah 2:2). That is, he appears to have been not only in the sea, but also "in the belly of the earth" (Matthew 12:40). Yona indicates that he had descended not to the depths of the sea, but to the "roots of the mountains" (Jonah 2:6), which are the far interior recesses of the earth.
The Hebrew word for Sheol is the same word used for the destination of the humanity of the Person of Jesus Christ (Psalm 16:10 <=> Acts 2:27 and Acts 2:31). That is, the Septuagint of the Hebrew Bible uses the same word "Hades" in Psalm 16:10 as is found in Acts 2:27 and Acts 2:31. The same Hebrew word "Sheol" occurs in Psalm 16:10 and Jonah 2:2. Thus Sheol = Hades.
In other words, Yona did NOT enter an earthly grave (but was in the belly of the great fish in the Mediterranean Sea), and of course Jesus was NOT buried at sea, but was laid to rest in an earthly tomb. Yet while one dead body was in the sea, and one dead body was in the tomb, they both were in Sheol/Hades. That is, Sheol/Hades was the destination of the souls of the dead in the Old Testament.
Finally, at not least, Yona indicates that the Lord brought his life from "the pit" (Jonah 2:6). (His "life" here was not his "nefesh" [soul] but his "Che" [body of life].) In the Septuagint this word for "the pit" is the same root word used in the NT in the context of the "corruption" (or decay) of the body of Jesus Christ, which did not happen (Acts 2:27 and Acts 2:31). In other words, the parallel between the physical death of Yona and the physical death of Jesus was not only that they were both dead for three days and three nights (and that they both went to Sheol/Hades "in the belly of the earth") but that neither of their bodies saw "corruption."