= Solution
Here “contractive” means <strictly distance-decreasing>: $\boxed{d(Tx,Ty)<d(x,y)\text{ whenever }x\ne y}$. “Strongly contractive” means a <contraction mapping> with one uniform constant $q\in[0,1)$ satisfying $\boxed{d(Tx,Ty)\leq qd(x,y)\text{ for all }x,y\in M}$.
If distinct $p,q$ were both <fixed points>, strict decrease would give $d(p,q)=d(Tp,Tq)<d(p,q)$, a contradiction. Thus there is \b[at most one fixed point]. This proves uniqueness if a fixed point exists; the pointwise strict inequality alone does not assert existence.
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