The main result of this article reads as follows: if \(X\) is a separable Banach space and \(S\) is the unit sphere of an almost locally uniformly rotund norm on \(X\), any continuous map on \(S\) with values in an arbitrary Banach space is approximable within \(\epsilon\) by a \(C^n\)-smooth function on some open neighbourhood \(U\) of \(S\), where \(n\) is an arbitrary integer and \(U\) depends upon \(\epsilon\). This theorem should be compared with a result shown in [\textit{J. Frontisi}, Rocky Mt. J. Math. 25, 1295--1304 (1995; Zbl 0853.46015)] which states: if a Banach space \(X\) has an equivalent locally uniformly rotund norm and if every norm on \(X\) is a uniform limit on bounded sets of \(C^k\)-smooth functions, then the space \(X\) has \(C^k\)-smooth partitions of unity.