Constructions are given of different kinds of flats in the projective space PG(9,2)=\mathbb P(Ù²V(5,2))PG(9,2)={\mathbb P}(\wedge^{2}V(5,2)) which are external to the Grassmannian G_1,4,2{\cal G}_{\bf 1,4,2} of lines of PG(4,2). In particular it is shown that there exist precisely two GL(5,2)-orbits of external 4-flats, each with stabilizer group 31:5. (No 5-flat is external.) For each k=1,2,3, two distinct kinds of external k-flats are simply constructed out of certain partial spreads in PG(4,2) of size k+2. A third kind of external plane, with stabilizer 2³:(7:3), is also shown to exist. With the aid of a certain key counting lemma, it is proved that the foregoing amounts to a complete classification of external flats.