  In this paper we generalise the notion of Drinfeld modular form for the group
$\Gamma$ := GL2(Fq[$\theta$]) to a vector-valued setting, where the target
spaces are certain modules over positive characteristic Banach algebras over
which are defined what we call the 'representations of the first kind'. Under
quite reasonable restrictions, we show that the spaces of such modular forms
are finite-dimensional, are endowed with certain generalisations of Hecke
operators, with differential operators{\`a} la Serre etc. The crucial point of
this work is the introduction of a 'field of uniformisers', a valued field in
which we can study the expansions at the cusp infinity of our modular forms and
which is wildly ramified. Examples of such modular forms are given through the
construction of Poincar{\'e} and Eisenstein series. After this the paper
continues with a more detailed analysis of the special case of modular forms
associated to a restricted class of representations $\rho$ * $\Sigma$ of
$\Gamma$ which has more importance in arithmetical applications. More structure
results are given in this case, and a harmonic product formula is obtained
which allows, with the help of three conjectures on the structure of an
Fp-algebra of A-periodic multiple sums, to produce conjectural formulas for
Eisenstein series. Some of these formulas can be proved.
