(Joint work with Brian Conrad.) Let  K  be a complete rank 1 valued field with ring of integers  OK, A an adic noetherian ring and  f:A→OK  an adic morphism. If  g:X→Y  is a proper flat morphism between rigid analytic spaces over  K  then locally on  Y  a flat formal model of g spreads out to a proper flat morphism between formal schemes topologically of finite type over  A. As an application one can prove that for proper smooth  g  and  K  of characteristic 0, the Hodge to de Rham spectral sequence for g degenerates and the  Rqg？ΩpX/Y  are locally free. 

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