$p$-Clique is W[1]-complete under fpt-reductions [FG06, Theorems 6.1 and 6.35]: it is in W[1] ($WH_D01_CliqueInW1$), and it is W[1]-hard since it is A[1]-hard ($WH_D06_CliqueA1Complete$) and W[1] = A[1] ($WH_D09_W1EqA1$).

A problem is W[1]-hard as soon as Clique, or any other W[1]-hard problem, fpt-reduces to it ($WH_A3_ReductionCalculus.hard_of_fptReduces$); it is then not fixed-parameter tractable unless $\mathrm{W}[1] \subseteq \mathrm{FPT}$ ($WH_B5_HierarchyFacts.not_mem_FPT_of_hard$).