No — "Specialist" in the U.S. Army is a single enlisted rank (E-4), not a graded series that goes up to a "Specialist 5." The historical "Specialist 5" (SP5) rank did exist, but it was retired decades ago.
Here's the background: from the 1950s through the 1980s, the U.S. Army used a parallel "Specialist" rank track (SP4 through SP9, later trimmed to SP4–SP7) to give technically skilled soldiers pay grades comparable to noncommissioned officers (corporals, sergeants, etc.) without requiring them to hold leadership/command authority. Specialist 5 (SP5), paired with pay grade E-5, was equivalent in pay to a sergeant but carried no NCO leadership responsibilities. This system was phased out by 1985 as the Army consolidated its enlisted structure, eliminating all Specialist grades above SP4.
Today, "Specialist" refers only to pay grade E-4, ranking above Private First Class (E-3) and below Corporal/Sergeant (E-5). Modern soldiers who reach pay grade E-5 are simply called Sergeants, since the Specialist track above E-4 no longer exists. So if someone asks about "Specialist 5," they're either referring to a historical rank from before 1985, or possibly confusing it with a different branch's structure (some other U.S. military and government systems, like the Cyber and Space Force enlisted tracks or certain federal civilian classifications, use different numbering conventions, but these aren't Army enlisted ranks).
In short: current active-duty Army rank structure has no "Specialist 5" — only "Specialist" (E-4) exists in that family today, with Sergeant (E-5) being the next grade up.