Coordination number of carbon in methyllithium tetramer
TL; DR: coordination number of carbon atom in methyllithium tetramer is indeed 7, arising from 6 intramolecular interactions, as you suggested, and an extra bond with the lithium atom of the next tetramer.



The key point is that you need to look beyond the tetramer. I'm sorry to inform you that the image from Wikipedia of the standalone cubane $\ce{Li4(CH3)4}$ is useless for the determination of coordination number (C.N.) of carbon. In fact, there is a 3D network of interlinked clusters, and the description given in J. D. Lee's Concise Inorganic Chemistry is in fact very precise [1, pp 304]:


  The structure of the $\ce{(LiCH3)4}$ cluster is unusual. The four $\ce{Li}$ atoms occupy the corners of a tetrahedron. Each methyl $\ce{C}$ atom is above a face of the tetrahedron, and forms a triple bridge to the three $\ce{Li}$ atoms that make up the face of the tetrahedron. The intramolecular $\ce{Li-C}$ distance is 
  $2.31~\mathrm{Å}$. The $\ce{C}$ is bonded to the three $\ce{H}$ atoms in the methyl group. The $\ce{C}$ is also bonded to a $\ce{Li}$ atom in another tetrahedron (with an intermolecular $\ce{Li-C}$ distance of $2.36~\mathrm{Å}$). The coordination number for the $\ce{C}$ atom is therefore $7$.


The problem I stumbled upon is that I didn't find any ready-to-use CIFs for methyl lithium with 3D coordinates (there are plenty for the derivatives and some powder XRD of questionable quality). I ended up creating my own CIF based on data published by Weiss et al. [2] with VESTA program and visualized the content of the unit cell with OLEX2: 



Figure 1. Unit cell of the methyl lithium tetramer with the central cubane core $\ce{Li4(CH3)4}$ linked via four crystallographically equivalent $\ce{C}$ atoms with 4 neighboring cubanes displayed as tetrahedra $\ce{Li4}$ for clarity. Color code: $\color{#EEEEEE}{\Large\bullet}~\ce{H}$; $\color{#CC80FF}{\Large\bullet}~\ce{Li}$; $\color{#909090}{\Large\bullet}~\ce{C}$.



Figure 2. Coordination environment of the carbon atom and interatomic distances (Å). 

In addition, I run the test for C.N. with Dirichlet subroutine of the ToposPro, which also suggests the C.N. 7:

Central atom:C1 0.320 0.320 0.320 Rsd:0.900
 D(CP):0.624  ( 0.3698 0.3698 0.3698 )
 D(VDP):0.364  ( 0.3490 0.3490 0.3490 )
Atom:0.980 &lt; r &lt; 2.370  &lt;r&gt;=1.747   Top: 1.229 &lt; R &lt; 1.556  &lt;R&gt;=1.357
CN=7:0:0 NV=10 V=3.052/6.126 S=13.376 Cpac=0.161 Ccov=5.175
G3=0.114818789
Face distribution: {3/1 4/3 5/3 }
Vertex distribution: {3/10 }

  N  Atom   x      y      z     Dist.   D1      D2     SAng. 
  1  H_1  0.3510  0.1920  0.3510  0.97957  0.89982  0.07975  27.31224
  2  H_1  0.1920  0.3510  0.3510  0.97957  0.89982  0.07975  27.31224
  3  H_1  0.3510  0.3510  0.1920  0.97957  0.89982  0.07975  27.31224
  4  Li1  0.3690  0.3690  0.6310  2.30686  0.89982  1.40704   5.76860
  5  Li1  0.3690  0.6310  0.3690  2.30686  0.89982  1.40704   5.76860
  6  Li1  0.6310  0.3690  0.3690  2.30686  0.89982  1.40704   5.76860
  7  Li1  0.1310  0.1310  0.1310  2.37007  0.89982  1.47025   0.75748




Figure 3. Voronoi-Dirichlet coordination polyhedron for the carbon atom.

Content of the CIF file:

#======================================================================

# CRYSTAL DATA

#----------------------------------------------------------------------

data_VESTA_phase_1


_chemical_name_common                  'Methyl lithium tetramer'
_cell_length_a                         7.24000
_cell_length_b                         7.24000
_cell_length_c                         7.24000
_cell_angle_alpha                      90
_cell_angle_beta                       90
_cell_angle_gamma                      90
_space_group_name_H-M_alt              'I -4 3 m'
_space_group_IT_number                 217

loop_
_space_group_symop_operation_xyz
   'x, y, z'
   '-x, -y, z'
   '-x, y, -z'
   'x, -y, -z'
   'z, x, y'
   'z, -x, -y'
   '-z, -x, y'
   '-z, x, -y'
   'y, z, x'
   '-y, z, -x'
   'y, -z, -x'
   '-y, -z, x'
   'y, x, z'
   '-y, -x, z'
   'y, -x, -z'
   '-y, x, -z'
   'x, z, y'
   '-x, z, -y'
   '-x, -z, y'
   'x, -z, -y'
   'z, y, x'
   'z, -y, -x'
   '-z, y, -x'
   '-z, -y, x'
   'x+1/2, y+1/2, z+1/2'
   '-x+1/2, -y+1/2, z+1/2'
   '-x+1/2, y+1/2, -z+1/2'
   'x+1/2, -y+1/2, -z+1/2'
   'z+1/2, x+1/2, y+1/2'
   'z+1/2, -x+1/2, -y+1/2'
   '-z+1/2, -x+1/2, y+1/2'
   '-z+1/2, x+1/2, -y+1/2'
   'y+1/2, z+1/2, x+1/2'
   '-y+1/2, z+1/2, -x+1/2'
   'y+1/2, -z+1/2, -x+1/2'
   '-y+1/2, -z+1/2, x+1/2'
   'y+1/2, x+1/2, z+1/2'
   '-y+1/2, -x+1/2, z+1/2'
   'y+1/2, -x+1/2, -z+1/2'
   '-y+1/2, x+1/2, -z+1/2'
   'x+1/2, z+1/2, y+1/2'
   '-x+1/2, z+1/2, -y+1/2'
   '-x+1/2, -z+1/2, y+1/2'
   'x+1/2, -z+1/2, -y+1/2'
   'z+1/2, y+1/2, x+1/2'
   'z+1/2, -y+1/2, -x+1/2'
   '-z+1/2, y+1/2, -x+1/2'
   '-z+1/2, -y+1/2, x+1/2'

loop_
   _atom_site_label
   _atom_site_occupancy
   _atom_site_fract_x
   _atom_site_fract_y
   _atom_site_fract_z
   _atom_site_adp_type
   _atom_site_B_iso_or_equiv
   _atom_site_type_symbol
   Li1        1.0     0.131000      0.131000      0.131000     Biso  1.000000 Li
   C1         1.0     0.320000      0.320000      0.320000     Biso  1.000000 C
   H1         1.0     0.351000      0.351000      0.192000     Biso  1.000000 H


References


Lee, J. D. Concise Inorganic Chemistry, 4th ed.; Chapman &amp; Hall: London ; New York, 1991. ISBN 978-0-412-40290-6.  
Weiss, E.; Hencken, G. Über Metall-Alkyl-Verbindungen: XII. Verfeinerung der Kristallstruktur des Methyllithiums. Journal of Organometallic Chemistry 1970, 21 (2), 265–268. DOI: 10.1016/S0022-328X(00)83621-9. (In German)