A burn is not necessary for a gravity assist maneuver. Oftentimes, just the flyby gives enough of an assist to get to either the destination or the next gravity assist. Without the burn, the planet can add velocity to the spacecraft while it changes the direction to a more desired direction.
https://en.wikipedia.org/wiki/Gravity_assist#Explanation

However, a powered flyby can be used, and the close approach flyby gives more oomph to the burn. This type of gravity assist uses what is called the Oberth effect, a magnification of the result of the burn. It is a more efficient use of the available propellant.
https://en.wikipedia.org/wiki/Oberth_effect

In the non-burn case, the assist comes from the motion of the planet. From the point of view of anyone on the planet, the spacecraft approached and left at the same speed on a hyperbolic orbital trajectory (the speed is greater than escape velocity), but the spacecraft leaves in a different direction than it approached. From the Sun’s point of view, the spacecraft has changed its speed along with its direction, and this is the effect the mission planners wanted, a change in the orbit around the Sun so that the spacecraft can get to the next planet on its tour of the solar system. In the case of the burn, even more change in speed can be gained and a slightly different escape direction would result.

—I’m having a Biden Moment…