so far I too haven't come across of any such report where the motif alone can act as a functional unit> As you say, if it does so then it is against the definition of motif and hence it can not be called as motif.

There are quite a lot of interaction motifs that are based on short sequence patterns, called SLiM (short linear motif) or ELM (eukaryotic linear motif), and they are commonly employed by intrinsically disordered proteins to interact with their targets. See database link enclosed. 

And yes it is possible to call them domain, see e.g. the D-domain that interacts with the D-site recognition site on the MAP kinase ERK2. 

http://bioware.ucd.ie/~compass/biowareweb//

Motif and domain as useful concepts but always mean different things in my view. A motif can be structural, such as a helix-hairpin-hexlix motif, or functional, such as a PCNA interacting peptide (PIP) motif. The first crystal structures for a given class of enzyme may for example identify functional motifs that then can be used to examine sequences for possible similar activities and structural motifs that can be seen in patterns in other sequences that will form a three-dimensional arrangement of secondary structure. A domain is a ideally an independent folding unit and minimally a unit with more interactions within it than with other regions, so that in might be stable as a separate unit. Consider Jane Richardson's classifications of protein structures, which are domain based - e.g. The Anatomy and Taxonomy of Protein Structure - see http://kinemage.biochem.duke.edu/teaching/anatax/index.html.

Domains are in fact compact three-dimensional structures that typically have independent stability and folding, which is why Jane Richardson could group them into classes and NMR researchers can break proteins into domains to solve their structures by divide-and-conquer strategies.  On the other hand, as noted by Ruth, linear peptide motifs can often act as independent functional units as in blocking an interaction. Motifs may be able to evolve and function independently from the rest of the protein, two features of domains as noted by Wikipedia.  However,  I still would not call these linear sequences  domains since they may lack any internal interactions and are typically not independent folding units or units with internal structural integrity. Phosphorylation may cause a recognition of a motif by FHA domains or BRCT domains, which bind phosphorylated sequences, but these domains are not motifs and the recognized phosphorylation motif is not a domain. Domains are typically 100 to 300 residues and larger protein sequences often form multiple linked folded domains, so a domain is in essence always a basic building block for protein structures, whereas a motif is usually not. A structural motif is typically a supersecondary structure, as in a helix-hairpin-helix, which may occur in different types of domains. A functional sequence motif is an amino-acid sequence pattern that lacks most features of a domain, although for example Walker A and Walker B motifs acting in ATP binding are protein sequence motifs that have conserved three-dimensional structures in the context of their folded protein domains.  I hope this may help your thinking. 

Ruth and John provide definitive observations. But it occurred to me on reading them that one of the outstanding features (functions?) of the eucaryotic genome is the ability to experience sequence duplications and crossovers, both of which are usually larger than the typical small sequence of a motif, but would allow for usage of established motif's in new protein domains. If you chose to label that as a "function" then, yes, motifs would have that additional function. Still, they are most often embedded in a larger domain on which they depend for folding stability (or sometimes they themselves add more stability to the more "global" folding) so it would be hard to qualify them as independent functional units given so much dependency, except on that epigenetic level just suggested. To further confuse the situation, we in molecular genetics must recognize that currently in complexity systems science the word "motif" has a very specific meeting within a network structure. To wit, the answers here presume the subdiscipline of molecular genetics not network theory. However, isomorphically the reasoning for the molecular answers also seem spot on for the network instantiation.