Not with current technology. Most space missions get their initial boost at the beginning via launch and orbital escape burns. They may pick up a little speed via slingshot around planets, stealing a bit of momentum. However, while five probes (to date) have gone fast enough that they will never "fall back" into the sun, the sun's gravity still reduces their speed over time.
At least until the probe crosses into the region where the target star's gravity dominates. Then it will accelerate as it approaches the star. Same thing happened with the Apollo missions. The spacecraft gradually slowed down until it passed the Earth-Moon equilibrium point, and then started to speed up again.
Now technically a plutonium-powered ion engine could produce a near-continuous thrust for quite a while that would keep accelerating the spacecraft. However, there hasn't yet been enough development to keep such an engine active long enough to shave significant time off the trip.
This link says the fastest probe will go 450Kmph - I don’t know how credible it is..
https://astronomy.com/news/2021/04/warp-drives-physicists-investigate-faster-than-light-space-travel