Suffering in Gehenna is not physical, but spiritual. When the soul yearns to join with its heavenly source but cannot yet because of its sins, it must undergo afflictions.

Punishment in Gehenna is not physical:

What is this judgment called Gehenna? If you say that the punishment comes to a person’s body after his death, his body is as lifeless as a stone, and whether you burn his bones into lime or anoint them with balsam [oil]…which preserves them, it is of no difference to the stone. This is the same with regard to the body of a righteous person and that of a wicked person after their deaths. Furthermore, how could this body be in Gehenna, but that one in the good of the World to Come, when they are buried before you in one grave or concealed in a chamber in one coffin? Rather, the punishment is entirely spiritual.

The soul yearns to ascend and connect to holiness, and it is afflicted when it is unable to do so:

The punishment of Gehenna comes to a person immediately after death. As soon as the wicked person dies, his soul is tied to a wheel of fire, and it then joins the river of fire that emerges from beneath the heavenly throne of glory. The river descends to Gehenna, and the soul descends with it…In its essence, this soul, which returns to the element of fire and is attached to it, longs to ascend and to connect to the heavens, but the coarseness of the individual’s sins separates [the soul] from its Creator, preventing this from occurring. Instead, the soul is pulled into the fire of Gehenna and joined to it. Having this wish denied is an affliction that brings immeasurable suffering, in addition to the suffering of Gehenna. This is what is meant by the excision of the soul; it is cut off from its root like a branch that is chopped from a tree. (Ramban, Torat HaAdam, Sha’ar HaGemul)

The Sages called the suffering of sinners in Gehenna “the pocket of the slingshot.” The soul is propelled in two different directions: toward the desires it became accustomed to in the [physical] world, which it is now unable to satisfy; and toward its yearning to connect to God, which it has not yet learned to do.

After death, the soul of the sinner longs for spiritual elevation but is unable to attain it:

This is the anguish and punishment of the soul: When a person is alive, he pursues his desires and physical concerns, and his soul is distracted from doing God’s will and becomes accustomed to acting in accordance with the body’s inclinations, which are the opposite of its own natural inclinations. When the soul separates from the body, it yearns for the things it was accustomed to, but it does not possess the means to attain them [because it no longer has a body]. By its own nature, it is inclined to embrace heavenly pursuits, which are [entirely] different from physical ones, and it yearns for them, but it does not have the basic ability, the education, or the experience of serving God, and someone who has not prepared and trained his soul for this cannot attain that pleasure.… Consequently, the soul yearns for two opposites simultaneously, the high and the low, one as a result of [the soul’s] nature and the other as a result of [its] experience. It does not possess the means to attain the lower, nor the preparation to attain the higher.

The “pocket of the slingshot” refers to the affliction of the soul that yearns for two opposites:

This causes suffering that is more painful than any suffering in the [physical] world; more than any kind of broken bone, than being burned by fire, than the terrible suffering of the freezing cold, than being attacked with knives and swords, and than being bitten by snakes and scorpions…This is as our Sages said in tractate Shabbat (153b), “Rabbi Eliezer says, ‘The souls of the righteous are preserved beneath the heavenly throne of glory…but for those of the wicked…one angel stands at one end of the world and another angel stands at the other end of the world and they sling the souls from one to the other, as it is stated, “The souls of your enemies may He cast away as from the pocket of a slingshot” (I Samuel 25:29).’” This alludes to [the soul’s] two opposing desires. (Rav Yosef Albo, Sefer HaIkkarim 4:33)

The sinner damages the root of his soul and strengthens the forces of impurity. While still alive, he is already in a state of Gehenna, which surrounds him while he sins.

With regard to the punishment of Gehenna, the sin itself is the sinner’s punishment…. When a person does something that is prohibited by one of the negative commandments, the [resulting] damage and destruction is immediately inscribed above, heaven forbid, in the root of his [soul]…. He [thereby] raises and strengthens the forces of impurity and spiritual negativity, may God protect us, from which he draws a spirit of impurity upon himself, and this enwraps him while he performs the sin. After he has performed the sin, the spirit of impurity returns to its place. As such during his lifetime, he is truly in Gehenna, which surrounds him when he sins, but he does not feel it until after he dies, when he will be trapped in the net that he has prepared: [A net] comprised of the forces of impurity and the demons that were created by his own actions. (Rabbi Ĥayyim of Volozhin, Nefesh HaĤayyim 1:12)