Let Y be a closed subspace of a Banach space X. An element \(z\in Y\) is called a strongly unique best approximation of order \(\alpha\) \((\alpha >1)\) at x, if for some \(M>0\) there exists \(\gamma =\gamma (x,M)>0\) such that, for all \(y\in Y\) with \(\| y-z\| \leq M\), \(\| y-x\| \geq \| z-x\| +\gamma \| y-z\|^{\alpha}.\) Results concerning strong uniqueness in this sense are given for spaces whose modulus of convexity or smoothness is of ``power type p'' for some \(p>0\). Other results are concerned with strongly unique best approximation with respect to a strictly increasing function \(\alpha\) from \(R^+\) to \(R^+\) with \(\alpha (0)=0\). These results partly extend those given by \textit{B. Prus} and \textit{R. Smarzewski} [J. Math. Anal. Appl. 121, 10-21 (1987; Zbl 0617.41046)].